{"id":1830,"date":"2016-03-17T18:18:36","date_gmt":"2016-03-17T18:18:36","guid":{"rendered":"https:\/\/courses.candelalearning.com\/nrocarithmetic\/?post_type=chapter&#038;p=1830"},"modified":"2026-02-20T21:41:31","modified_gmt":"2026-02-20T21:41:31","slug":"rules-for-adding-real-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/rules-for-adding-real-numbers\/","title":{"raw":"Real Numbers","rendered":"Real Numbers"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Add and subtract real numbers\r\n<ul>\r\n \t<li>Add\u00a0real numbers with the same and different signs<\/li>\r\n \t<li>Subtract real numbers with the same and different signs<\/li>\r\n \t<li>Simplify combinations that require both addition and subtraction of real numbers.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Multiply and divide real numbers\r\n<ul>\r\n \t<li>Multiply two or more real numbers.<\/li>\r\n \t<li>Divide real numbers<\/li>\r\n \t<li>Simplify expressions with both multiplication and division<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Simplify compound expressions with real numbers\r\n<ul>\r\n \t<li>Use the Order of Operations to simplify an expression<\/li>\r\n \t<li>Simplify expressions containing exponents<\/li>\r\n \t<li>Simplify expressions with multiple grouping symbols<\/li>\r\n \t<li>Simplify expressions containing absolute values<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\nSome important terminology to remember before we begin is as follows:\r\n<ul>\r\n \t<li><strong>integers:\u00a0<\/strong>counting numbers like 1, 2, 3, etc., including negatives and zero<\/li>\r\n \t<li><strong>real number:\u00a0<\/strong>fractions, negative numbers, decimals, integers, and zero are all real numbers<\/li>\r\n \t<li><strong>absolute value:<\/strong> a number's distance from zero; it's always positive. \u00a0[latex]|-7| = 7[\/latex]<\/li>\r\n \t<li><strong>sign:\u00a0<\/strong>this refers to whether a number is positive or negative, we use [latex]+[\/latex] for positive (to the right of zero on the number line) and [latex]-[\/latex] for negative (to the left of zero on the number line)<\/li>\r\n \t<li><strong>difference:\u00a0<\/strong>the result of subtraction<\/li>\r\n \t<li><strong>sum:\u00a0<\/strong>the result of addition<\/li>\r\n<\/ul>\r\nThe ability to work comfortably with negative numbers is essential to success in algebra. For this reason we will do a quick review of adding, subtracting, multiplying and dividing integers. Integers are all the positive whole numbers, zero, and their opposites (negatives). As this is intended to be a review of integers, the descriptions and examples will not be as detailed as a normal lesson.\r\n<h2>Add and subtract real numbers<\/h2>\r\n<h3>Add real numbers<\/h3>\r\nIn this section, we will use the skills from the last section to simplify mathematical expressions that contain many grouping symbols and many operations. We are using the term compound to describe expressions that have many operations and many grouping symbols. More care is needed with these expressions when you apply the order of operations. Additionally, you will see how to handle absolute value terms when you simplify expressions.\r\n\r\nWhen adding integers we have two cases to consider. The first case is whether\u00a0the signs match (both positive or both negative). If the signs match, we will add the numbers together and keep the sign.\r\n<div class=\"textbox exercises\">\r\n<h3>Example 1<\/h3>\r\n<ol>\r\n \t<li>Add 21 + 34<\/li>\r\n \t<li>Add -18 + (-13)<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"588555\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"588555\"]\r\n\r\n1. Because both numbers are positive, we add 21 and 34 and the result is also positive.\r\n<p style=\"text-align: center;\">21 + 34 = 55<\/p>\r\n<p style=\"text-align: left;\">2.\u00a0Because both numbers are negative, we add 18 and 13 and the result is negative.<\/p>\r\n<p style=\"text-align: center;\">-18 + (-13) = - 31<\/p>\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIf the signs don\u2019t match (one positive and one negative number) we will subtract the numbers (as if they were all positive) and then use the sign from the larger number. This means if the larger number is positive, the answer is positive. If the larger number is negative, the answer is negative.\r\n<div class=\"textbox exercises\">\r\n<h3>Example 2<\/h3>\r\n<ol>\r\n \t<li>Add [latex]-10+6[\/latex]<\/li>\r\n \t<li>Add [latex]12+(- 4)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"363546\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"363546\"]\r\n\r\n1. Because the numbers are different signs, we subtract the absolute values and keep the sign of the number with the larger absolute value.\r\n<ul>\r\n \t<li>Subtracting the absolute values gives the result of 4.\u00a0 \u00a0 \u00a0[latex]10-6=4[\/latex]<\/li>\r\n \t<li style=\"text-align: left;\">Since the 10 is negative, the result is negative.\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]-10+6=-4[\/latex]<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n2.\u00a0Because the numbers are different signs, we subtract the absolute values and keep the sign of the number with the larger absolute value.\r\n<ul>\r\n \t<li>Subtracting the absolute values gives the result of 8.\u00a0 \u00a0 \u00a0\u00a0[latex]12-4=8[\/latex]<\/li>\r\n \t<li>Since the 12 is positive, the result is positive.\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]12+(-4)=8[\/latex]<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h3>To add two numbers with the same sign (both positive or both negative)<\/h3>\r\n<ul>\r\n \t<li><i>Add<\/i> their absolute values (without the [latex]+[\/latex] or [latex]-[\/latex] sign)<\/li>\r\n \t<li>Give the sum the same sign.<\/li>\r\n<\/ul>\r\n<h3>To add two numbers with different signs (one positive and one negative)<\/h3>\r\n<ul>\r\n \t<li>Find the<i> difference <\/i>oftheir absolute values. (Note that when you find the difference of the absolute values, you always subtract the lesser absolute value from the greater one.)<\/li>\r\n \t<li>Give the sum the same sign as the number with the greater absolute value.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Subtract real numbers<\/h3>\r\nSubtraction can be defined as \"adding the opposite.\"\u00a0 In the following example, we will rewrite the subtraction problem as adding the opposite and apply what we learned about adding real numbers to simplify.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 3<\/h3>\r\nFind\u00a0[latex]23\u201373[\/latex].\r\n\r\n[reveal-answer q=\"951238\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"951238\"]You can't use your usual method of subtraction because 73 is greater than 23.\u00a0Rewrite the subtraction as adding the opposite.\r\n<p style=\"text-align: center;\">[latex]23+\\left(\u221273\\right)[\/latex]<\/p>\r\nThe addends have different signs, so find the difference of their absolute values.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left|23\\right|=23\\,\\,\\,\\text{and}\\,\\,\\,\\left|\u221273\\right|=73\\\\73-23=50\\end{array}[\/latex]<\/p>\r\nSince [latex]\\left|\u221273\\right|&gt;\\left|23\\right|[\/latex], the final answer is negative.\r\n<h4>Answer<\/h4>\r\n[latex]23\u201373=\u221250[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAnother way to think about subtracting is to think about the distance between the two numbers on the number line. In the example below, [latex]382[\/latex] is to the <i>right<\/i> of 0 by [latex]382[\/latex] units, and [latex]\u221293[\/latex] is to the <i>left<\/i> of 0 by 93 units. The distance between them is the sum of their distances to 0: [latex]382+93[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170754\/image050.gif\" alt=\"A number line from negative 93 to 382. Negative 93 is 93 units from 0 and 382 is 382 units from 0. The total distance from negative 93 to 382 can be found by adding them together. 382+93=475 units.\" width=\"575\" height=\"147\" \/>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 4<\/h3>\r\nFind\u00a0[latex]382\u2013\\left(\u221293\\right)[\/latex].\r\n\r\n[reveal-answer q=\"342295\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"342295\"]You are subtracting a negative, so think of this as taking the negative sign away. This becomes an addition problem. [latex]-93[\/latex] becomes [latex]+93[\/latex]\r\n<p style=\"text-align: center;\">[latex]382+93=475[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]382\u2013(\u221293)=475[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video explains how to subtract two signed integers.\r\n\r\nhttps:\/\/youtu.be\/ciuIKFCtWWU\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 5<\/h3>\r\nFind [latex]-\\frac{3}{7}-\\frac{6}{7}+\\frac{2}{7}[\/latex]\r\n\r\n[reveal-answer q=\"11416\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"11416\"]Add the first two and give the result a negative sign:\r\n\r\nSince the signs of the first two are the same, find the sum of the absolute values of the fractions\r\n\r\nSince both numbers are negative, the sum is negative. If you owe money, then borrow more, the amount you owe becomes larger.\r\n<p style=\"text-align: center;\">[latex]\\left| -\\frac{3}{7} \\right|=\\frac{3}{7}[\/latex] and [latex]\\left| -\\frac{6}{7} \\right|=\\frac{6}{7}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{3}{7}+\\frac{6}{7}=\\frac{9}{7}\\\\\\\\-\\frac{3}{7}-\\frac{6}{7} =-\\frac{9}{7}\\end{array}[\/latex]<\/p>\r\nNow add the third number. The signs are different, so find the <em>difference<\/em> of their absolute values.\r\n<p style=\"text-align: center;\">[latex] \\left| -\\frac{9}{7} \\right|=\\frac{9}{7}[\/latex] and [latex] \\left| \\frac{2}{7} \\right|=\\frac{2}{7}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\frac{9}{7}-\\frac{2}{7}=\\frac{7}{7}[\/latex]<\/p>\r\nSince [latex]\\left|\\frac{-9}{7}\\right|&gt;\\left|\\frac{2}{7}\\right|[\/latex], the sign of the final sum is the same as the sign of [latex]-\\frac{9}{7}[\/latex].<i>\r\n<\/i>\r\n<p style=\"text-align: center;\">[latex] -\\frac{9}{7}+\\frac{2}{7}=-\\frac{7}{7} = -1[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]-\\frac{3}{7}+\\left(-\\frac{6}{7}\\right)+\\frac{2}{7}=-\\frac{7}{7} = -1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video you will see an example of how to add three fractions with a common denominator that have different signs.\r\n\r\nhttps:\/\/youtu.be\/P972VVbR98k\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 6<\/h3>\r\nEvaluate [latex]27.832+(\u22123.06)[\/latex]. When you add decimals, remember to line up the decimal points so you are adding tenths to tenths, hundredths to hundredths, and so on.\r\n\r\n[reveal-answer q=\"545871\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"545871\"]Since the addends have different signs, subtract their absolute values.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}\\underline{\\begin{array}{r}27.832\\\\-\\text{ }3.06\\,\\,\\,\\end{array}}\\\\24.772\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left|-3.06\\right|=3.06[\/latex]<\/p>\r\nThe sum has the same sign as 27.832 whose absolute value is greater.\r\n<h4>Answer<\/h4>\r\n[latex]27.832+\\left(-3.06\\right)=24.772[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video are\u00a0examples of adding and subtracting decimals with different signs.\r\n\r\nhttps:\/\/youtu.be\/3FHZQ5iKcpI\r\n<h2>Multiply and divide real numbers<\/h2>\r\nMultiplication and division are <strong>inverse operations<\/strong>, just as addition and subtraction are. You may recall that when you divide fractions, you multiply by the reciprocal. Inverse operations \"undo\" each other.\r\n<h3>Multiply real numbers<\/h3>\r\nMultiplying real numbers is not that different from multiplying whole numbers and positive fractions. However, you haven't learned what effect a negative sign has on the product.\r\n\r\nWith whole numbers, you can think of multiplication as repeated addition. Using the number line, you can make multiple jumps of a given size. For example, the following picture shows the product [latex]3\\cdot4[\/latex] as 3 jumps of 4 units each.\r\n\r\n<img id=\"Picture 262\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170800\/image054.gif\" alt=\"A number line showing 3 times 4 is 12. From the 0, the right-facing person jumps 4 units at a time, and jumps 3 times. The person lands on 12.\" width=\"521\" height=\"130\" \/>\r\n\r\nSo to multiply [latex]3(\u22124)[\/latex], you can face left (toward the negative side) and make three \u201cjumps\u201d forward (in a negative direction).\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170801\/image055.jpg\" alt=\"A number line representing 3 times negative 4 equals negative 12. A left-facing person jumps left 4 spaces 3 times so that the person lands on negative 12.\" width=\"516\" height=\"136\" \/>\r\n\r\nThe product of a positive number and a negative number (or a negative and a positive) is negative.\r\n<div class=\"textbox shaded\">\r\n<h3>The Product of a Positive Number and a Negative Number<\/h3>\r\nTo multiply a <strong>positive number<\/strong> and a <strong>negative number<\/strong>, multiply their absolute values. The product is <strong>negative<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 7<\/h3>\r\nFind [latex]\u22123.8(0.6)[\/latex].\r\n[reveal-answer q=\"456211\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"456211\"]Multiply the absolute values as you normally would.\u00a0Place the decimal point by counting place values.\u00a03.8 has 1 place after the decimal point, and 0.6 has 1 place after the decimal point, so the product has [latex]1+1[\/latex] or 2 places after the decimal point.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}3.8\\\\\\underline{\\times\\,\\,\\,0.6}\\\\2.28\\end{array}[\/latex]<\/p>\r\nThe product of a negative and a positive is negative.\r\n<h4>Answer<\/h4>\r\n[latex]\u22123.8(0.6)=\u22122.28[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video contains examples of how to multiply decimal numbers with different signs.\r\n\r\nhttps:\/\/youtu.be\/7gY0S3LUUyQ\r\n<div class=\"textbox shaded\">\r\n<h3>The Product of Two Numbers with the Same Sign (both positive or both negative)<\/h3>\r\nTo multiply two <strong>positive numbers<\/strong>, multiply their absolute values. The product is <strong>positive<\/strong>.\r\n\r\nTo multiply two <strong>negative numbers<\/strong>, multiply their absolute values. The product is <strong>positive<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 8<\/h3>\r\nFind [latex] ~\\left( -\\frac{3}{4} \\right)\\left( -\\frac{2}{5} \\right)[\/latex]\r\n\r\n[reveal-answer q=\"322816\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"322816\"]Multiply the absolute values of the numbers.\u00a0First, multiply the numerators together to get the product's numerator. Then, multiply the denominators together to get the product's denominator. Rewrite in lowest terms, if needed.\r\n<p style=\"text-align: center;\">[latex] \\left( \\frac{3}{4} \\right)\\left( \\frac{2}{5} \\right)=\\frac{6}{20}=\\frac{3}{10}[\/latex]<\/p>\r\nThe product of two negative numbers is positive.\r\n<h4>Answer<\/h4>\r\n[latex] \\left( -\\frac{3}{4} \\right)\\left( -\\frac{2}{5} \\right)=\\frac{3}{10}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows examples of multiplying two signed fractions, including simplification of the answer.\r\n\r\nhttps:\/\/youtu.be\/yUdJ46pTblo\r\n\r\nTo summarize:\r\n<ul>\r\n \t<li><strong>positive <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0positive<\/i>:<\/strong> The product is <strong>positive<\/strong>.<\/li>\r\n \t<li><strong>negative <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0negative<\/i>:<\/strong> The product is <strong>positive<\/strong>.<\/li>\r\n \t<li><strong>negative <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0positive<\/i>:<\/strong> The product is <strong>negative<\/strong>.<\/li>\r\n \t<li><strong>positive <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0negative<\/i>:<\/strong> The product is <strong>negative<\/strong>.<\/li>\r\n<\/ul>\r\nYou can see that the product of two negative numbers is a positive number. So, if you are multiplying more than two numbers, you can count the number of negative factors.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplying More Than Two Negative Numbers<\/h3>\r\nIf there are an <strong>even<\/strong> number (0, 2, 4, ...) of negative factors to multiply, the product is <strong>positive<\/strong>.\r\nIf there are an <strong>odd<\/strong> number (1, 3, 5, ...) of negative factors, the product is <strong>negative<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 9<\/h3>\r\nFind [latex]3(\u22126)(2)(\u22123)(\u22121)[\/latex].\r\n\r\n[reveal-answer q=\"149062\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"149062\"]Multiply the absolute values of the numbers.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}3(6)(2)(3)(1)\\\\18(2)(3)(1)\\\\36(3)(1)\\\\108(1)\\\\108\\end{array}[\/latex]<\/p>\r\nCount the number of negative factors. There are three [latex]\\left(\u22126,\u22123,\u22121\\right)[\/latex].\r\n<p style=\"text-align: center;\">[latex]3(\u22126)(2)(\u22123)(\u22121)[\/latex]<\/p>\r\nSince there are an odd number of negative factors, the product is negative.\r\n<h4>Answer<\/h4>\r\n[latex]3(\u22126)(2)(\u22123)(\u22121)=\u2212108[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video contains examples of multiplying more than two signed integers.\r\n\r\nhttps:\/\/youtu.be\/rx8F9SPd0HE\r\n<h3>Divide real numbers<\/h3>\r\nYou may remember that when you divided fractions, you multiplied by the <strong>reciprocal<\/strong>. <i>Reciprocal <\/i>is another name for the multiplicative inverse (just as <i>opposite <\/i>is another name for additive inverse).\r\n\r\nAn easy way to find the multiplicative inverse is to just \u201cflip\u201d the numerator and denominator as you did to find the reciprocal. Here are some examples:\r\n<ul>\r\n \t<li>The reciprocal of [latex]\\frac{4}{9}[\/latex]\u00a0is [latex] \\frac{9}{4}[\/latex]because [latex]\\frac{4}{9}\\left(\\frac{9}{4}\\right)=\\frac{36}{36}=1[\/latex].<\/li>\r\n \t<li>The reciprocal of 3 is [latex]\\frac{1}{3}[\/latex]\u00a0because [latex]\\frac{3}{1}\\left(\\frac{1}{3}\\right)=\\frac{3}{3}=1[\/latex].<\/li>\r\n \t<li>The reciprocal of [latex]-\\frac{5}{6}[\/latex]\u00a0is [latex]\\frac{-6}{5}[\/latex]\u00a0because [latex]-\\frac{5}{6}\\left( -\\frac{6}{5} \\right)=\\frac{30}{30}=1[\/latex].<\/li>\r\n \t<li>The reciprocal of 1 is 1 as [latex]1(1)=1[\/latex].<\/li>\r\n<\/ul>\r\nWhen you divided by positive fractions, you learned to multiply by the reciprocal. You also do this to divide <strong>real numbers<\/strong>.\r\n\r\nThink about dividing a bag of 26 marbles into two smaller bags with the same number of marbles in each. You can also say each smaller bag has <i>one half<\/i> of the marbles.\r\n<p style=\"text-align: center;\">[latex] 26\\div 2=26\\left( \\frac{1}{2} \\right)=13[\/latex]<\/p>\r\nNotice that 2 and [latex] \\frac{1}{2}[\/latex] are reciprocals.\r\n\r\nTry again, dividing a bag of 36 marbles into smaller bags.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>Number of bags<\/th>\r\n<th>Dividing by number of bags<\/th>\r\n<th>Multiplying by reciprocal<\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<thead><\/thead>\r\n<tbody>\r\n<tr>\r\n<td>3<\/td>\r\n<td>[latex]\\frac{36}{3}=12[\/latex]<\/td>\r\n<td>[latex] 36\\left( \\frac{1}{3} \\right)=\\frac{36}{3}=\\frac{12(3)}{3}=12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>[latex]\\frac{36}{4}=9[\/latex]<\/td>\r\n<td>[latex]36\\left(\\frac{1}{4}\\right)=\\frac{36}{4}=\\frac{9\\left(4\\right)}{4}=9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6<\/td>\r\n<td>[latex]\\frac{36}{6}=6[\/latex]<\/td>\r\n<td>[latex]36\\left(\\frac{1}{6}\\right)=\\frac{36}{6}=\\frac{6\\left(6\\right)}{6}=6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDividing by a number is the same as multiplying by its reciprocal. (That is, you use the reciprocal of the <strong>divisor<\/strong>, the second number in the division problem.)\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 10<\/h3>\r\nFind [latex] 28\\div \\frac{4}{3}[\/latex]\r\n\r\n[reveal-answer q=\"210216\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"210216\"]Rewrite the division as multiplication by the reciprocal. The reciprocal of [latex] \\frac{4}{3}[\/latex] is [latex]\\frac{3}{4} [\/latex].\r\n<p style=\"text-align: center;\">[latex] 28\\div \\frac{4}{3}=28\\left( \\frac{3}{4} \\right)[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex]\\frac{28}{1}\\left(\\frac{3}{4}\\right)=\\frac{28\\left(3\\right)}{4}=\\frac{4\\left(7\\right)\\left(3\\right)}{4}=7\\left(3\\right)=21[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]28\\div\\frac{4}{3}=21[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow let's see what this means when one or more of the numbers is negative. A number and its reciprocal have the same sign. Since division is rewritten as multiplication using the reciprocal of the divisor, and taking the reciprocal doesn\u2019t change any of the signs, division follows the same rules as multiplication.\r\n<div class=\"textbox shaded\">\r\n<h3>Rules of Division<\/h3>\r\nWhen dividing, rewrite the problem as multiplication using the reciprocal of the divisor as the second factor.\r\n\r\nWhen one number is <strong>positive<\/strong> and the other is <strong>negative<\/strong>, the <strong>quotient<\/strong> is <strong>negative<\/strong>.\r\n\r\nWhen <em>both<\/em> numbers are <strong>negative<\/strong>, the quotient is <strong>positive<\/strong>.\r\n\r\nWhen <em>both<\/em> numbers are <strong>positive<\/strong>, the quotient is <strong>positive<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 11<\/h3>\r\nFind [latex]24\\div\\left(-\\frac{5}{6}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"716581\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"716581\"]Rewrite the division as multiplication by the reciprocal.\r\n<p style=\"text-align: center;\">[latex] 24\\div \\left( -\\frac{5}{6} \\right)=24\\left( -\\frac{6}{5} \\right)[\/latex]<\/p>\r\nMultiply. Since one number is positive and one is negative, the product is negative.\r\n<p style=\"text-align: center;\">[latex] \\frac{24}{1}\\left( -\\frac{6}{5} \\right)=-\\frac{144}{5}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] 24\\div \\left( -\\frac{5}{6} \\right)=-\\frac{144}{5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 12<\/h3>\r\nFind [latex] 4\\,\\left( -\\frac{2}{3} \\right)\\,\\div \\left( -6 \\right)[\/latex]\r\n\r\n[reveal-answer q=\"557653\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"557653\"]Rewrite the division as multiplication by the reciprocal.\r\n<p style=\"text-align: center;\">[latex] \\frac{4}{1}\\left( -\\frac{2}{3} \\right)\\left( -\\frac{1}{6} \\right)[\/latex]<\/p>\r\nMultiply. There is an even number of negative numbers, so the product is positive.\r\n<p style=\"text-align: center;\">[latex]\\frac{4\\left(2\\right)\\left(1\\right)}{3\\left(6\\right)}=\\frac{8}{18}[\/latex]<\/p>\r\nWrite the fraction in lowest terms.\r\n<h4>Answer<\/h4>\r\n[latex] 4\\left( -\\frac{2}{3} \\right)\\div \\left( -6 \\right)=\\frac{4}{9}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video explains how to divide signed fractions.\r\n\r\nhttps:\/\/youtu.be\/OPHdadhDJoI\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"traffic-sign-160659\" width=\"103\" height=\"91\" \/>\r\n\r\nRemember that a fraction bar also indicates division, so a negative sign in front of a fraction goes with the numerator, the denominator, or the whole fraction: [latex]-\\frac{3}{4}=\\frac{-3}{4}=\\frac{3}{-4}[\/latex].\r\n\r\nIn each case, the overall fraction is negative because there's only one negative in the division.\r\n\r\n<\/div>\r\nThe following video explains how to divide signed fractions.\r\n\r\nhttps:\/\/youtu.be\/OPHdadhDJoI\r\n<h2>Simplify compound expressions with real numbers<\/h2>\r\nIn this section, we will simplify mathematical expressions that contain many grouping symbols and many operations. We are using the term compound to describe expressions that have many operations and sometimes many grouping symbols. Additionally, you will see how to handle absolute value terms when you simplify expressions.\r\n<h3>Use the order of operations to simplify an expression<\/h3>\r\nYou may or may not recall the order of operations for applying several mathematical operations to one expression. Just as it is a social convention for us to drive on the right-hand side of the road, the order of operations is a set of conventions used to provide order when you are required to use several mathematical operations for one expression. The graphic below depicts the order in which mathematical operations are performed.\r\n\r\n[caption id=\"attachment_4963\" align=\"aligncenter\" width=\"756\"]<img class=\" wp-image-4963\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/17181059\/Screen-Shot-2016-06-17-at-10.57.52-AM-300x183.png\" alt=\"steps of order of operations that say Perform all operations within grouping symbols first. Grouping symbols include {}, [], () Evaluate exponents or square roots Multiply or divide from left to right Add or subtract from left to right\" width=\"756\" height=\"461\" \/> Order of operations[\/caption]\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 13<\/h3>\r\nSimplify [latex]7\u20135+3\\cdot8[\/latex].\r\n\r\n[reveal-answer q=\"987816\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"987816\"]According to the order of operations, multiplication comes before addition and subtraction.\r\n\r\nMultiply [latex]3\\cdot8[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}7\u20135+3\\cdot8\\\\7\u20135+24\\end{array}[\/latex]<\/p>\r\nNow, add and subtract from left to right. [latex]7\u20135[\/latex] comes first.\r\n<p style=\"text-align: center;\">[latex]2+24[\/latex].<\/p>\r\n<p style=\"text-align: left;\">Finally, add.<\/p>\r\n<p style=\"text-align: center;\">[latex]2+24=26[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]7\u20135+3\\cdot8=26[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following example, you will be shown how to simplify an expression that contains both multiplication and subtraction using the order of operations.\r\n\r\nhttps:\/\/youtu.be\/yFO_0dlfy-w\r\n\r\nWhen you are applying the order of operations to expressions that contain fractions, decimals, and negative numbers, you will need to recall how to do these computations as well.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 14<\/h3>\r\nSimplify [latex]3\\cdot\\frac{1}{3}-8\\div\\frac{1}{4}[\/latex].\r\n\r\n[reveal-answer q=\"265256\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"265256\"]According to the order of operations, multiplication and division come before addition and subtraction. Sometimes it helps to add parentheses to help you know what comes first, so let's put parentheses around the multiplication and division since it will come before the subtraction.\r\n<p style=\"text-align: center;\">[latex]3\\cdot\\frac{1}{3}-8\\div\\frac{1}{4}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Multiply [latex] 3\\cdot \\frac{1}{3}[\/latex] first.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left(3\\cdot\\frac{1}{3}\\right)-\\left(8\\div\\frac{1}{4}\\right)\\\\\\text{}\\\\=\\left(1\\right)-\\left(8\\div \\frac{1}{4}\\right)\\end{array}[\/latex]<\/p>\r\nNow, divide [latex]8\\div\\frac{1}{4}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}8\\div\\frac{1}{4}=\\frac{8}{1}\\cdot\\frac{4}{1}=32\\\\\\text{}\\\\1-32\\end{array}[\/latex]<\/p>\r\nSubtract.\r\n<p style=\"text-align: center;\">[latex]1\u201332=\u221231[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] 3\\cdot \\frac{1}{3}-8\\div \\frac{1}{4}=-31[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video you are shown how to use the order of operations to simplify an expression that contains multiplication, division, and subtraction with terms that contain fractions.\r\n\r\nhttps:\/\/youtu.be\/yqp06obmcVc\r\n<h3>Simplify expressions containing exponents<\/h3>\r\nWhen you are evaluating expressions, you will sometimes see exponents used to represent repeated multiplication. Recall that an expression such as [latex]7^{2}[\/latex]\u00a0is <strong>exponential notation<\/strong> for [latex]7\\cdot7[\/latex]. (Exponential notation has two parts: the <strong>base<\/strong> and the <strong>exponent<\/strong> or the <strong>power<\/strong>. In [latex]7^{2}[\/latex], 7 is the base and 2 is the exponent; the exponent determines how many times the base is multiplied by itself.)\r\n\r\nExponents are a way to represent repeated multiplication; the order of operations places it <i>before <\/i>any other multiplication, division, subtraction, and addition is performed.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 15<\/h3>\r\nSimplify [latex]3^{2}\\cdot2^{3}[\/latex].\r\n\r\n[reveal-answer q=\"360237\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"360237\"]This problem has exponents and multiplication in it. According to the order of operations, simplifying\u00a0[latex]3^{2}[\/latex]\u00a0and [latex]2^{3}[\/latex]\u00a0comes before multiplication.\r\n<p style=\"text-align: center;\">[latex]3^{2}\\cdot2^{3}[\/latex]<\/p>\r\n[latex] {{3}^{2}}[\/latex] is [latex]3\\cdot3[\/latex], which equals 9.\r\n<p style=\"text-align: center;\">[latex] 9\\cdot {{2}^{3}}[\/latex]<\/p>\r\n[latex] {{2}^{3}}[\/latex] is [latex]2\\cdot2\\cdot2[\/latex], which equals 8.\r\n<p style=\"text-align: center;\">[latex] 9\\cdot 8[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex] 9\\cdot 8=72[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] {{3}^{2}}\\cdot {{2}^{3}}=72[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the video that follows, an expression with exponents on its terms is simplified using the order of operations.\r\n\r\nhttps:\/\/youtu.be\/JjBBgV7G_Qw\r\n<h3>Simplify expressions containing multiple grouping symbols<\/h3>\r\nGrouping symbols such as parentheses ( ), brackets [ ], braces[latex] \\displaystyle \\left\\{ {} \\right\\}[\/latex], and fraction bars can be used to further control the order of the four arithmetic operations.\u00a0The rules of the order of operations require computation within grouping symbols to be completed first, even if you are adding or subtracting within the grouping symbols and you have multiplication outside the grouping symbols. After computing within the grouping symbols, divide or multiply from left to right and then subtract or add from left to right.\u00a0When there are grouping symbols within grouping symbols, calculate from the inside to the outside. That is, begin simplifying within the innermost grouping symbols first.\r\n\r\nRemember that parentheses can also be used to show multiplication. In the example that follows, both uses of parentheses\u2014as a way to represent a group, as well as a way to express multiplication\u2014are shown.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 16<\/h3>\r\nSimplify [latex]\\left(3+4\\right)^{2}+\\left(8\\right)\\left(4\\right)[\/latex].\r\n\r\n[reveal-answer q=\"548490\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"548490\"]This problem has parentheses, exponents, multiplication, and addition in it. The first set of parentheses is a grouping symbol. The second set indicates multiplication.\r\n\r\nGrouping symbols are handled first. Add numbers in parentheses.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}(3+4)^{2}+(8)(4)\\\\(7)^{2}+(8)(4)\\end{array}[\/latex]<\/p>\r\nSimplify\u00a0[latex]7^{2}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}7^{2}+(8)(4)\\\\49+(8)(4)\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Multiply.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}49+(8)(4)\\\\49+(32)\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Add.<\/p>\r\n<p style=\"text-align: center;\">[latex]49+32=81[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex](3+4)^{2}+(8)(4)=81[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 17<\/h3>\r\nSimplify \u00a0[latex]4\\cdot{\\frac{3[5+{(2 + 3)}^2]}{2}}[\/latex]\r\n[reveal-answer q=\"358226\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"358226\"]\r\n\r\nThere are brackets and parentheses in this problem. Compute inside the innermost grouping symbols first.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}4\\cdot{\\frac{3[5+{(2 + 3)}^2]}{2}}\\\\\\text{ }\\\\=4\\cdot{\\frac{3[5+{(5)}^2]}{2}}\\end{array}[\/latex]<\/p>\r\nThen apply the exponent\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}4\\cdot{\\frac{3[5+{(5)}^2]}{2}}\\\\\\text{}\\\\=4\\cdot{\\frac{3[5+25]}{2}}\\\\\\text{ }\\\\=4\\cdot{\\frac{3[30]}{2}}\\end{array}[\/latex]<\/p>\r\nThen simplify the fraction\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}4\\cdot{\\frac{3[30]}{2}}\\\\\\text{}\\\\=4\\cdot{\\frac{90}{2}}\\\\\\text{ }\\\\=4\\cdot{45}\\\\\\text{ }\\\\=180\\end{array}[\/latex]<\/p>\r\n\r\n<h4 style=\"text-align: left;\">Answer<\/h4>\r\n[latex]4\\cdot{\\frac{3[5+{(2 + 3)}^2]}{2}}=180[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you are shown how to use the order of operations to simplify an expression with grouping symbols, exponents, multiplication, and addition.\r\n\r\nhttps:\/\/youtu.be\/EMch2MKCVdA\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nThese problems are very similar to the examples given above. How are they different and what tools do you need to simplify them?\r\n\r\na) Simplify\u00a0[latex]\\left(1.5+3.5\\right)\u20132\\left(0.5\\cdot6\\right)^{2}[\/latex].\u00a0This problem has parentheses, exponents, multiplication, subtraction, and addition in it, as well as decimals instead of integers.\r\n\r\nUse the box below to write down a few thoughts about how you would simplify this expression with decimals and grouping symbols.\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n<p class=\"p1\">[reveal-answer q=\"680970\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"680970\"]\r\nGrouping symbols are handled first. Add numbers in the first set of parentheses.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}(1.5+3.5)\u20132(0.5\\cdot6)^{2}\\\\5\u20132(0.5\\cdot6)^{2}\\end{array}[\/latex]<\/p>\r\nMultiply numbers in the second set of parentheses.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}5\u20132(0.5\\cdot6)^{2}\\\\5\u20132(3)^{2}\\end{array}[\/latex]<\/p>\r\nEvaluate exponents.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}5\u20132(3)^{2}\\\\5\u20132\\cdot9\\end{array}[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}5\u20132\\cdot9\\\\5\u201318\\end{array}[\/latex]<\/p>\r\nSubtract.\r\n<p style=\"text-align: center;\">[latex]5\u201318=\u221213[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex](1.5+3.5)\u20132(0.5\\cdot6)^{2}=\u221213[\/latex]\r\n\r\n[\/hidden-answer]\r\n<p class=\"p1\">b) Simplify [latex] {{\\left( \\frac{1}{2} \\right)}^{2}}+{{\\left( \\frac{1}{4} \\right)}^{3}}\\cdot \\,32[\/latex].<\/p>\r\nUse the box below to write down a few thoughts about how you would simplify this expression with fractions and grouping symbols.\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n[reveal-answer q=\"680972\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"680972\"]\r\nThis problem has exponents, multiplication, and addition in it, as well as fractions instead of integers.\r\n\r\nAccording to the order of operations, simplify the terms with the exponents first, then multiply, then add.\r\n<p style=\"text-align: center;\">[latex]\\left(\\frac{1}{2}\\right)^{2}+\\left(\\frac{1}{4}\\right)^{3}\\cdot32[\/latex]<\/p>\r\nEvaluate: [latex]\\left(\\frac{1}{2}\\right)^{2}=\\frac{1}{2}\\cdot\\frac{1}{2}=\\frac{1}{4}[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\frac{1}{4}+\\left(\\frac{1}{4}\\right)^{3}\\cdot32[\/latex]<\/p>\r\nEvaluate: [latex]\\left(\\frac{1}{4}\\right)^{3}=\\frac{1}{4}\\cdot\\frac{1}{4}\\cdot\\frac{1}{4}=\\frac{1}{64}[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\frac{1}{4}+\\frac{1}{64}\\cdot32[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex] \\frac{1}{4}+\\frac{32}{64}[\/latex]<\/p>\r\nSimplify. [latex] \\frac{32}{64}=\\frac{1}{2}[\/latex], so you can add [latex] \\frac{1}{4}+\\frac{1}{2}[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\frac{1}{4}+\\frac{1}{2}=\\frac{3}{4}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] {{\\left( \\frac{1}{2} \\right)}^{2}}+{{\\left( \\frac{1}{4} \\right)}^{3}}\\cdot 32=\\frac{3}{4}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 18<\/h3>\r\nSimplify [latex] \\frac{5-[3+(2\\cdot (-6))]}{{{3}^{2}}+2}[\/latex]\r\n\r\n[reveal-answer q=\"906386\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"906386\"]This problem has brackets, parentheses, fractions, exponents, multiplication, subtraction, and addition in it.\r\n\r\nGrouping symbols are handled first. The parentheses around the [latex]-6[\/latex] aren\u2019t a grouping symbol; they are simply making it clear that the negative sign belongs to the 6. Start with the innermost set of parentheses that are a grouping symbol. In this example, the innermost set of parentheses\u00a0would be in\u00a0the numerator of the fraction, [latex](2\\cdot(\u22126))[\/latex]. Begin working out from there. (The fraction line acts as a type of grouping symbol, too; you simplify the numerator and denominator independently, and then divide the numerator by the denominator at the end.)\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{5-\\left[3+\\left(2\\cdot\\left(-6\\right)\\right)\\right]}{3^{2}+2}\\\\\\\\\\frac{5-\\left[3+\\left(-12\\right)\\right]}{3^{2}+2}\\end{array}[\/latex]<\/p>\r\nAdd [latex]3[\/latex] and [latex]-12[\/latex], which are in brackets, to get [latex]-9[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{5-\\left[3+\\left(-12\\right)\\right]}{3^{2}+2}\\\\\\\\\\frac{5-\\left[-9\\right]}{3^{2}+2}\\end{array}[\/latex]<\/p>\r\nSubtract [latex]5\u2013\\left[\u22129\\right]=5+9=14[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{5-\\left[-9\\right]}{3^{2}+2}\\\\\\\\\\frac{14}{3^{2}+2}\\end{array}[\/latex]<\/p>\r\nThe top of the fraction is all set, but the bottom (denominator) has remained untouched. Apply the order of operations to that as well. Begin by evaluating\u00a0[latex]3^{2}=9[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{14}{3^{2}+2}\\\\\\\\\\frac{14}{9+2}\\end{array}[\/latex]<\/p>\r\nNow add. [latex]9+2=11[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{14}{9+2}\\\\\\\\\\frac{14}{11}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{5-\\left[3+\\left(2\\cdot\\left(-6\\right)\\right)\\right]}{3^{2}+2}=\\frac{14}{11}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe video that follows contains an example similar to the written one above. Note how the numerator and denominator of the fraction are simplified separately.\r\n\r\nhttps:\/\/youtu.be\/xIJLq54jM44\r\n\r\n&nbsp;\r\n<h3>Simplify expressions containing absolute values<\/h3>\r\nAbsolute value expressions are one final method of grouping that you may see. Recall that the absolute value of a quantity is always positive or 0.\r\n\r\nWhen you see an absolute value expression included within a larger expression, treat the absolute value like a grouping symbol and evaluate the expression within the absolute value sign first. Then take the absolute value of that expression. The example below shows how this is done.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example 19<\/h3>\r\nSimplify [latex]\\frac{3+\\left|2-6\\right|}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}[\/latex].\r\n\r\n[reveal-answer q=\"572632\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"572632\"]This problem has absolute values, decimals, multiplication, subtraction, and addition in it.\r\n\r\nGrouping symbols, including absolute value, are handled first. Simplify the numerator, then the denominator.\r\n\r\nEvaluate [latex]\\left|2\u20136\\right|[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{3+\\left|2-6\\right|}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\\\\\\\\\frac{3+\\left|-4\\right|}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\end{array}[\/latex]<\/p>\r\nTake the absolute value of [latex]\\left|\u22124\\right|[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{3+\\left|-4\\right|}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\\\\\\\\\frac{3+4}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\end{array}[\/latex]<\/p>\r\nAdd the numbers in the numerator.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{3+4}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\\\\\\\\\frac{7}{2\\left| 3\\cdot 1.5 \\right|-(-3)}\\end{array}[\/latex]<\/p>\r\nNow that the numerator is simplified, turn to the denominator.\r\n\r\nEvaluate the absolute value expression first. [latex]3 \\cdot 1.5 = 4.5[\/latex], giving\r\n<p style=\"text-align: center;\">\u00a0[latex]\\begin{array}{c}\\frac{7}{2\\left|{3\\cdot{1.5}}\\right|-(-3)}\\\\\\\\\\frac{7}{2\\left|{ 4.5}\\right|-(-3)}\\end{array}[\/latex]<\/p>\r\nThe expression \u201c[latex]2\\left|4.5\\right|[\/latex]\u201d reads \u201c2 times the absolute value of 4.5.\u201d Multiply 2 times 4.5.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{7}{2\\left|4.5\\right|-\\left(-3\\right)}\\\\\\\\\\frac{7}{9-\\left(-3\\right)}\\end{array}[\/latex]<\/p>\r\nSubtract.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{7}{9-\\left(-3\\right)}\\\\\\\\\\frac{7}{12}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{3+\\left|2-6\\right|}{2\\left|3\\cdot1.5\\right|-3\\left(-3\\right)}=\\frac{7}{12}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe following video uses the order of operations to simplify an expression in fraction form that contains absolute value terms. Note how the absolute values are treated like parentheses and brackets when using the order of operations.\r\n\r\nhttps:\/\/youtu.be\/6wmCQprxlnU","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Add and subtract real numbers\n<ul>\n<li>Add\u00a0real numbers with the same and different signs<\/li>\n<li>Subtract real numbers with the same and different signs<\/li>\n<li>Simplify combinations that require both addition and subtraction of real numbers.<\/li>\n<\/ul>\n<\/li>\n<li>Multiply and divide real numbers\n<ul>\n<li>Multiply two or more real numbers.<\/li>\n<li>Divide real numbers<\/li>\n<li>Simplify expressions with both multiplication and division<\/li>\n<\/ul>\n<\/li>\n<li>Simplify compound expressions with real numbers\n<ul>\n<li>Use the Order of Operations to simplify an expression<\/li>\n<li>Simplify expressions containing exponents<\/li>\n<li>Simplify expressions with multiple grouping symbols<\/li>\n<li>Simplify expressions containing absolute values<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>Some important terminology to remember before we begin is as follows:<\/p>\n<ul>\n<li><strong>integers:\u00a0<\/strong>counting numbers like 1, 2, 3, etc., including negatives and zero<\/li>\n<li><strong>real number:\u00a0<\/strong>fractions, negative numbers, decimals, integers, and zero are all real numbers<\/li>\n<li><strong>absolute value:<\/strong> a number&#8217;s distance from zero; it&#8217;s always positive. \u00a0[latex]|-7| = 7[\/latex]<\/li>\n<li><strong>sign:\u00a0<\/strong>this refers to whether a number is positive or negative, we use [latex]+[\/latex] for positive (to the right of zero on the number line) and [latex]-[\/latex] for negative (to the left of zero on the number line)<\/li>\n<li><strong>difference:\u00a0<\/strong>the result of subtraction<\/li>\n<li><strong>sum:\u00a0<\/strong>the result of addition<\/li>\n<\/ul>\n<p>The ability to work comfortably with negative numbers is essential to success in algebra. For this reason we will do a quick review of adding, subtracting, multiplying and dividing integers. Integers are all the positive whole numbers, zero, and their opposites (negatives). As this is intended to be a review of integers, the descriptions and examples will not be as detailed as a normal lesson.<\/p>\n<h2>Add and subtract real numbers<\/h2>\n<h3>Add real numbers<\/h3>\n<p>In this section, we will use the skills from the last section to simplify mathematical expressions that contain many grouping symbols and many operations. We are using the term compound to describe expressions that have many operations and many grouping symbols. More care is needed with these expressions when you apply the order of operations. Additionally, you will see how to handle absolute value terms when you simplify expressions.<\/p>\n<p>When adding integers we have two cases to consider. The first case is whether\u00a0the signs match (both positive or both negative). If the signs match, we will add the numbers together and keep the sign.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example 1<\/h3>\n<ol>\n<li>Add 21 + 34<\/li>\n<li>Add -18 + (-13)<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q588555\">Show Answer<\/span><\/p>\n<div id=\"q588555\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. Because both numbers are positive, we add 21 and 34 and the result is also positive.<\/p>\n<p style=\"text-align: center;\">21 + 34 = 55<\/p>\n<p style=\"text-align: left;\">2.\u00a0Because both numbers are negative, we add 18 and 13 and the result is negative.<\/p>\n<p style=\"text-align: center;\">-18 + (-13) = &#8211; 31<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>If the signs don\u2019t match (one positive and one negative number) we will subtract the numbers (as if they were all positive) and then use the sign from the larger number. This means if the larger number is positive, the answer is positive. If the larger number is negative, the answer is negative.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example 2<\/h3>\n<ol>\n<li>Add [latex]-10+6[\/latex]<\/li>\n<li>Add [latex]12+(- 4)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q363546\">Show Answer<\/span><\/p>\n<div id=\"q363546\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. Because the numbers are different signs, we subtract the absolute values and keep the sign of the number with the larger absolute value.<\/p>\n<ul>\n<li>Subtracting the absolute values gives the result of 4.\u00a0 \u00a0 \u00a0[latex]10-6=4[\/latex]<\/li>\n<li style=\"text-align: left;\">Since the 10 is negative, the result is negative.\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]-10+6=-4[\/latex]<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>2.\u00a0Because the numbers are different signs, we subtract the absolute values and keep the sign of the number with the larger absolute value.<\/p>\n<ul>\n<li>Subtracting the absolute values gives the result of 8.\u00a0 \u00a0 \u00a0\u00a0[latex]12-4=8[\/latex]<\/li>\n<li>Since the 12 is positive, the result is positive.\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]12+(-4)=8[\/latex]<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h3>To add two numbers with the same sign (both positive or both negative)<\/h3>\n<ul>\n<li><i>Add<\/i> their absolute values (without the [latex]+[\/latex] or [latex]-[\/latex] sign)<\/li>\n<li>Give the sum the same sign.<\/li>\n<\/ul>\n<h3>To add two numbers with different signs (one positive and one negative)<\/h3>\n<ul>\n<li>Find the<i> difference <\/i>oftheir absolute values. (Note that when you find the difference of the absolute values, you always subtract the lesser absolute value from the greater one.)<\/li>\n<li>Give the sum the same sign as the number with the greater absolute value.<\/li>\n<\/ul>\n<\/div>\n<h3>Subtract real numbers<\/h3>\n<p>Subtraction can be defined as &#8220;adding the opposite.&#8221;\u00a0 In the following example, we will rewrite the subtraction problem as adding the opposite and apply what we learned about adding real numbers to simplify.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 3<\/h3>\n<p>Find\u00a0[latex]23\u201373[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q951238\">Show Solution<\/span><\/p>\n<div id=\"q951238\" class=\"hidden-answer\" style=\"display: none\">You can&#8217;t use your usual method of subtraction because 73 is greater than 23.\u00a0Rewrite the subtraction as adding the opposite.<\/p>\n<p style=\"text-align: center;\">[latex]23+\\left(\u221273\\right)[\/latex]<\/p>\n<p>The addends have different signs, so find the difference of their absolute values.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left|23\\right|=23\\,\\,\\,\\text{and}\\,\\,\\,\\left|\u221273\\right|=73\\\\73-23=50\\end{array}[\/latex]<\/p>\n<p>Since [latex]\\left|\u221273\\right|>\\left|23\\right|[\/latex], the final answer is negative.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]23\u201373=\u221250[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Another way to think about subtracting is to think about the distance between the two numbers on the number line. In the example below, [latex]382[\/latex] is to the <i>right<\/i> of 0 by [latex]382[\/latex] units, and [latex]\u221293[\/latex] is to the <i>left<\/i> of 0 by 93 units. The distance between them is the sum of their distances to 0: [latex]382+93[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170754\/image050.gif\" alt=\"A number line from negative 93 to 382. Negative 93 is 93 units from 0 and 382 is 382 units from 0. The total distance from negative 93 to 382 can be found by adding them together. 382+93=475 units.\" width=\"575\" height=\"147\" \/><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 4<\/h3>\n<p>Find\u00a0[latex]382\u2013\\left(\u221293\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q342295\">Show Solution<\/span><\/p>\n<div id=\"q342295\" class=\"hidden-answer\" style=\"display: none\">You are subtracting a negative, so think of this as taking the negative sign away. This becomes an addition problem. [latex]-93[\/latex] becomes [latex]+93[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]382+93=475[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]382\u2013(\u221293)=475[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video explains how to subtract two signed integers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 2:  Subtracting Integers (Two Digit Integers)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ciuIKFCtWWU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 5<\/h3>\n<p>Find [latex]-\\frac{3}{7}-\\frac{6}{7}+\\frac{2}{7}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q11416\">Show Solution<\/span><\/p>\n<div id=\"q11416\" class=\"hidden-answer\" style=\"display: none\">Add the first two and give the result a negative sign:<\/p>\n<p>Since the signs of the first two are the same, find the sum of the absolute values of the fractions<\/p>\n<p>Since both numbers are negative, the sum is negative. If you owe money, then borrow more, the amount you owe becomes larger.<\/p>\n<p style=\"text-align: center;\">[latex]\\left| -\\frac{3}{7} \\right|=\\frac{3}{7}[\/latex] and [latex]\\left| -\\frac{6}{7} \\right|=\\frac{6}{7}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{3}{7}+\\frac{6}{7}=\\frac{9}{7}\\\\\\\\-\\frac{3}{7}-\\frac{6}{7} =-\\frac{9}{7}\\end{array}[\/latex]<\/p>\n<p>Now add the third number. The signs are different, so find the <em>difference<\/em> of their absolute values.<\/p>\n<p style=\"text-align: center;\">[latex]\\left| -\\frac{9}{7} \\right|=\\frac{9}{7}[\/latex] and [latex]\\left| \\frac{2}{7} \\right|=\\frac{2}{7}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{9}{7}-\\frac{2}{7}=\\frac{7}{7}[\/latex]<\/p>\n<p>Since [latex]\\left|\\frac{-9}{7}\\right|>\\left|\\frac{2}{7}\\right|[\/latex], the sign of the final sum is the same as the sign of [latex]-\\frac{9}{7}[\/latex].<i><br \/>\n<\/i><\/p>\n<p style=\"text-align: center;\">[latex]-\\frac{9}{7}+\\frac{2}{7}=-\\frac{7}{7} = -1[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]-\\frac{3}{7}+\\left(-\\frac{6}{7}\\right)+\\frac{2}{7}=-\\frac{7}{7} = -1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video you will see an example of how to add three fractions with a common denominator that have different signs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Find the Sum and Difference of Three Signed Fractions (Common Denom)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/P972VVbR98k?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 6<\/h3>\n<p>Evaluate [latex]27.832+(\u22123.06)[\/latex]. When you add decimals, remember to line up the decimal points so you are adding tenths to tenths, hundredths to hundredths, and so on.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q545871\">Show Solution<\/span><\/p>\n<div id=\"q545871\" class=\"hidden-answer\" style=\"display: none\">Since the addends have different signs, subtract their absolute values.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underline{\\begin{array}{r}27.832\\\\-\\text{ }3.06\\,\\,\\,\\end{array}}\\\\24.772\\end{array}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\left|-3.06\\right|=3.06[\/latex]<\/p>\n<p>The sum has the same sign as 27.832 whose absolute value is greater.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]27.832+\\left(-3.06\\right)=24.772[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video are\u00a0examples of adding and subtracting decimals with different signs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Adding Signed Decimals\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/3FHZQ5iKcpI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Multiply and divide real numbers<\/h2>\n<p>Multiplication and division are <strong>inverse operations<\/strong>, just as addition and subtraction are. You may recall that when you divide fractions, you multiply by the reciprocal. Inverse operations &#8220;undo&#8221; each other.<\/p>\n<h3>Multiply real numbers<\/h3>\n<p>Multiplying real numbers is not that different from multiplying whole numbers and positive fractions. However, you haven&#8217;t learned what effect a negative sign has on the product.<\/p>\n<p>With whole numbers, you can think of multiplication as repeated addition. Using the number line, you can make multiple jumps of a given size. For example, the following picture shows the product [latex]3\\cdot4[\/latex] as 3 jumps of 4 units each.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" id=\"Picture 262\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170800\/image054.gif\" alt=\"A number line showing 3 times 4 is 12. From the 0, the right-facing person jumps 4 units at a time, and jumps 3 times. The person lands on 12.\" width=\"521\" height=\"130\" \/><\/p>\n<p>So to multiply [latex]3(\u22124)[\/latex], you can face left (toward the negative side) and make three \u201cjumps\u201d forward (in a negative direction).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170801\/image055.jpg\" alt=\"A number line representing 3 times negative 4 equals negative 12. A left-facing person jumps left 4 spaces 3 times so that the person lands on negative 12.\" width=\"516\" height=\"136\" \/><\/p>\n<p>The product of a positive number and a negative number (or a negative and a positive) is negative.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Product of a Positive Number and a Negative Number<\/h3>\n<p>To multiply a <strong>positive number<\/strong> and a <strong>negative number<\/strong>, multiply their absolute values. The product is <strong>negative<\/strong>.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 7<\/h3>\n<p>Find [latex]\u22123.8(0.6)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q456211\">Show Solution<\/span><\/p>\n<div id=\"q456211\" class=\"hidden-answer\" style=\"display: none\">Multiply the absolute values as you normally would.\u00a0Place the decimal point by counting place values.\u00a03.8 has 1 place after the decimal point, and 0.6 has 1 place after the decimal point, so the product has [latex]1+1[\/latex] or 2 places after the decimal point.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}3.8\\\\\\underline{\\times\\,\\,\\,0.6}\\\\2.28\\end{array}[\/latex]<\/p>\n<p>The product of a negative and a positive is negative.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\u22123.8(0.6)=\u22122.28[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video contains examples of how to multiply decimal numbers with different signs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex:  Multiplying Signed Decimals\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/7gY0S3LUUyQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox shaded\">\n<h3>The Product of Two Numbers with the Same Sign (both positive or both negative)<\/h3>\n<p>To multiply two <strong>positive numbers<\/strong>, multiply their absolute values. The product is <strong>positive<\/strong>.<\/p>\n<p>To multiply two <strong>negative numbers<\/strong>, multiply their absolute values. The product is <strong>positive<\/strong>.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 8<\/h3>\n<p>Find [latex]~\\left( -\\frac{3}{4} \\right)\\left( -\\frac{2}{5} \\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q322816\">Show Solution<\/span><\/p>\n<div id=\"q322816\" class=\"hidden-answer\" style=\"display: none\">Multiply the absolute values of the numbers.\u00a0First, multiply the numerators together to get the product&#8217;s numerator. Then, multiply the denominators together to get the product&#8217;s denominator. Rewrite in lowest terms, if needed.<\/p>\n<p style=\"text-align: center;\">[latex]\\left( \\frac{3}{4} \\right)\\left( \\frac{2}{5} \\right)=\\frac{6}{20}=\\frac{3}{10}[\/latex]<\/p>\n<p>The product of two negative numbers is positive.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left( -\\frac{3}{4} \\right)\\left( -\\frac{2}{5} \\right)=\\frac{3}{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows examples of multiplying two signed fractions, including simplification of the answer.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex:  Multiplying Signed Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/yUdJ46pTblo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>To summarize:<\/p>\n<ul>\n<li><strong>positive <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0positive<\/i>:<\/strong> The product is <strong>positive<\/strong>.<\/li>\n<li><strong>negative <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0negative<\/i>:<\/strong> The product is <strong>positive<\/strong>.<\/li>\n<li><strong>negative <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0positive<\/i>:<\/strong> The product is <strong>negative<\/strong>.<\/li>\n<li><strong>positive <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0negative<\/i>:<\/strong> The product is <strong>negative<\/strong>.<\/li>\n<\/ul>\n<p>You can see that the product of two negative numbers is a positive number. So, if you are multiplying more than two numbers, you can count the number of negative factors.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplying More Than Two Negative Numbers<\/h3>\n<p>If there are an <strong>even<\/strong> number (0, 2, 4, &#8230;) of negative factors to multiply, the product is <strong>positive<\/strong>.<br \/>\nIf there are an <strong>odd<\/strong> number (1, 3, 5, &#8230;) of negative factors, the product is <strong>negative<\/strong>.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 9<\/h3>\n<p>Find [latex]3(\u22126)(2)(\u22123)(\u22121)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q149062\">Show Solution<\/span><\/p>\n<div id=\"q149062\" class=\"hidden-answer\" style=\"display: none\">Multiply the absolute values of the numbers.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}3(6)(2)(3)(1)\\\\18(2)(3)(1)\\\\36(3)(1)\\\\108(1)\\\\108\\end{array}[\/latex]<\/p>\n<p>Count the number of negative factors. There are three [latex]\\left(\u22126,\u22123,\u22121\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]3(\u22126)(2)(\u22123)(\u22121)[\/latex]<\/p>\n<p>Since there are an odd number of negative factors, the product is negative.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3(\u22126)(2)(\u22123)(\u22121)=\u2212108[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video contains examples of multiplying more than two signed integers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Ex:  Multiplying Three or More Integers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/rx8F9SPd0HE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Divide real numbers<\/h3>\n<p>You may remember that when you divided fractions, you multiplied by the <strong>reciprocal<\/strong>. <i>Reciprocal <\/i>is another name for the multiplicative inverse (just as <i>opposite <\/i>is another name for additive inverse).<\/p>\n<p>An easy way to find the multiplicative inverse is to just \u201cflip\u201d the numerator and denominator as you did to find the reciprocal. Here are some examples:<\/p>\n<ul>\n<li>The reciprocal of [latex]\\frac{4}{9}[\/latex]\u00a0is [latex]\\frac{9}{4}[\/latex]because [latex]\\frac{4}{9}\\left(\\frac{9}{4}\\right)=\\frac{36}{36}=1[\/latex].<\/li>\n<li>The reciprocal of 3 is [latex]\\frac{1}{3}[\/latex]\u00a0because [latex]\\frac{3}{1}\\left(\\frac{1}{3}\\right)=\\frac{3}{3}=1[\/latex].<\/li>\n<li>The reciprocal of [latex]-\\frac{5}{6}[\/latex]\u00a0is [latex]\\frac{-6}{5}[\/latex]\u00a0because [latex]-\\frac{5}{6}\\left( -\\frac{6}{5} \\right)=\\frac{30}{30}=1[\/latex].<\/li>\n<li>The reciprocal of 1 is 1 as [latex]1(1)=1[\/latex].<\/li>\n<\/ul>\n<p>When you divided by positive fractions, you learned to multiply by the reciprocal. You also do this to divide <strong>real numbers<\/strong>.<\/p>\n<p>Think about dividing a bag of 26 marbles into two smaller bags with the same number of marbles in each. You can also say each smaller bag has <i>one half<\/i> of the marbles.<\/p>\n<p style=\"text-align: center;\">[latex]26\\div 2=26\\left( \\frac{1}{2} \\right)=13[\/latex]<\/p>\n<p>Notice that 2 and [latex]\\frac{1}{2}[\/latex] are reciprocals.<\/p>\n<p>Try again, dividing a bag of 36 marbles into smaller bags.<\/p>\n<table>\n<tbody>\n<tr>\n<th>Number of bags<\/th>\n<th>Dividing by number of bags<\/th>\n<th>Multiplying by reciprocal<\/th>\n<\/tr>\n<\/tbody>\n<thead><\/thead>\n<tbody>\n<tr>\n<td>3<\/td>\n<td>[latex]\\frac{36}{3}=12[\/latex]<\/td>\n<td>[latex]36\\left( \\frac{1}{3} \\right)=\\frac{36}{3}=\\frac{12(3)}{3}=12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>[latex]\\frac{36}{4}=9[\/latex]<\/td>\n<td>[latex]36\\left(\\frac{1}{4}\\right)=\\frac{36}{4}=\\frac{9\\left(4\\right)}{4}=9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>[latex]\\frac{36}{6}=6[\/latex]<\/td>\n<td>[latex]36\\left(\\frac{1}{6}\\right)=\\frac{36}{6}=\\frac{6\\left(6\\right)}{6}=6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Dividing by a number is the same as multiplying by its reciprocal. (That is, you use the reciprocal of the <strong>divisor<\/strong>, the second number in the division problem.)<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 10<\/h3>\n<p>Find [latex]28\\div \\frac{4}{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q210216\">Show Solution<\/span><\/p>\n<div id=\"q210216\" class=\"hidden-answer\" style=\"display: none\">Rewrite the division as multiplication by the reciprocal. The reciprocal of [latex]\\frac{4}{3}[\/latex] is [latex]\\frac{3}{4}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]28\\div \\frac{4}{3}=28\\left( \\frac{3}{4} \\right)[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{28}{1}\\left(\\frac{3}{4}\\right)=\\frac{28\\left(3\\right)}{4}=\\frac{4\\left(7\\right)\\left(3\\right)}{4}=7\\left(3\\right)=21[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]28\\div\\frac{4}{3}=21[\/latex]\n<\/p><\/div>\n<\/div>\n<\/div>\n<p>Now let&#8217;s see what this means when one or more of the numbers is negative. A number and its reciprocal have the same sign. Since division is rewritten as multiplication using the reciprocal of the divisor, and taking the reciprocal doesn\u2019t change any of the signs, division follows the same rules as multiplication.<\/p>\n<div class=\"textbox shaded\">\n<h3>Rules of Division<\/h3>\n<p>When dividing, rewrite the problem as multiplication using the reciprocal of the divisor as the second factor.<\/p>\n<p>When one number is <strong>positive<\/strong> and the other is <strong>negative<\/strong>, the <strong>quotient<\/strong> is <strong>negative<\/strong>.<\/p>\n<p>When <em>both<\/em> numbers are <strong>negative<\/strong>, the quotient is <strong>positive<\/strong>.<\/p>\n<p>When <em>both<\/em> numbers are <strong>positive<\/strong>, the quotient is <strong>positive<\/strong>.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 11<\/h3>\n<p>Find [latex]24\\div\\left(-\\frac{5}{6}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q716581\">Show Solution<\/span><\/p>\n<div id=\"q716581\" class=\"hidden-answer\" style=\"display: none\">Rewrite the division as multiplication by the reciprocal.<\/p>\n<p style=\"text-align: center;\">[latex]24\\div \\left( -\\frac{5}{6} \\right)=24\\left( -\\frac{6}{5} \\right)[\/latex]<\/p>\n<p>Multiply. Since one number is positive and one is negative, the product is negative.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{24}{1}\\left( -\\frac{6}{5} \\right)=-\\frac{144}{5}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]24\\div \\left( -\\frac{5}{6} \\right)=-\\frac{144}{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 12<\/h3>\n<p>Find [latex]4\\,\\left( -\\frac{2}{3} \\right)\\,\\div \\left( -6 \\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q557653\">Show Solution<\/span><\/p>\n<div id=\"q557653\" class=\"hidden-answer\" style=\"display: none\">Rewrite the division as multiplication by the reciprocal.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{4}{1}\\left( -\\frac{2}{3} \\right)\\left( -\\frac{1}{6} \\right)[\/latex]<\/p>\n<p>Multiply. There is an even number of negative numbers, so the product is positive.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{4\\left(2\\right)\\left(1\\right)}{3\\left(6\\right)}=\\frac{8}{18}[\/latex]<\/p>\n<p>Write the fraction in lowest terms.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]4\\left( -\\frac{2}{3} \\right)\\div \\left( -6 \\right)=\\frac{4}{9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video explains how to divide signed fractions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Ex 1:  Dividing Signed Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/OPHdadhDJoI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"traffic-sign-160659\" width=\"103\" height=\"91\" \/><\/p>\n<p>Remember that a fraction bar also indicates division, so a negative sign in front of a fraction goes with the numerator, the denominator, or the whole fraction: [latex]-\\frac{3}{4}=\\frac{-3}{4}=\\frac{3}{-4}[\/latex].<\/p>\n<p>In each case, the overall fraction is negative because there&#8217;s only one negative in the division.<\/p>\n<\/div>\n<p>The following video explains how to divide signed fractions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-8\" title=\"Ex 1:  Dividing Signed Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/OPHdadhDJoI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Simplify compound expressions with real numbers<\/h2>\n<p>In this section, we will simplify mathematical expressions that contain many grouping symbols and many operations. We are using the term compound to describe expressions that have many operations and sometimes many grouping symbols. Additionally, you will see how to handle absolute value terms when you simplify expressions.<\/p>\n<h3>Use the order of operations to simplify an expression<\/h3>\n<p>You may or may not recall the order of operations for applying several mathematical operations to one expression. Just as it is a social convention for us to drive on the right-hand side of the road, the order of operations is a set of conventions used to provide order when you are required to use several mathematical operations for one expression. The graphic below depicts the order in which mathematical operations are performed.<\/p>\n<div id=\"attachment_4963\" style=\"width: 766px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4963\" class=\"wp-image-4963\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/17181059\/Screen-Shot-2016-06-17-at-10.57.52-AM-300x183.png\" alt=\"steps of order of operations that say Perform all operations within grouping symbols first. Grouping symbols include {}, [], () Evaluate exponents or square roots Multiply or divide from left to right Add or subtract from left to right\" width=\"756\" height=\"461\" \/><\/p>\n<p id=\"caption-attachment-4963\" class=\"wp-caption-text\">Order of operations<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 13<\/h3>\n<p>Simplify [latex]7\u20135+3\\cdot8[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q987816\">Show Solution<\/span><\/p>\n<div id=\"q987816\" class=\"hidden-answer\" style=\"display: none\">According to the order of operations, multiplication comes before addition and subtraction.<\/p>\n<p>Multiply [latex]3\\cdot8[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}7\u20135+3\\cdot8\\\\7\u20135+24\\end{array}[\/latex]<\/p>\n<p>Now, add and subtract from left to right. [latex]7\u20135[\/latex] comes first.<\/p>\n<p style=\"text-align: center;\">[latex]2+24[\/latex].<\/p>\n<p style=\"text-align: left;\">Finally, add.<\/p>\n<p style=\"text-align: center;\">[latex]2+24=26[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]7\u20135+3\\cdot8=26[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following example, you will be shown how to simplify an expression that contains both multiplication and subtraction using the order of operations.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-9\" title=\"Simplify an Expression in the Form:  a-b+c*d\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/yFO_0dlfy-w?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>When you are applying the order of operations to expressions that contain fractions, decimals, and negative numbers, you will need to recall how to do these computations as well.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 14<\/h3>\n<p>Simplify [latex]3\\cdot\\frac{1}{3}-8\\div\\frac{1}{4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q265256\">Show Solution<\/span><\/p>\n<div id=\"q265256\" class=\"hidden-answer\" style=\"display: none\">According to the order of operations, multiplication and division come before addition and subtraction. Sometimes it helps to add parentheses to help you know what comes first, so let&#8217;s put parentheses around the multiplication and division since it will come before the subtraction.<\/p>\n<p style=\"text-align: center;\">[latex]3\\cdot\\frac{1}{3}-8\\div\\frac{1}{4}[\/latex]<\/p>\n<p style=\"text-align: left;\">Multiply [latex]3\\cdot \\frac{1}{3}[\/latex] first.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left(3\\cdot\\frac{1}{3}\\right)-\\left(8\\div\\frac{1}{4}\\right)\\\\\\text{}\\\\=\\left(1\\right)-\\left(8\\div \\frac{1}{4}\\right)\\end{array}[\/latex]<\/p>\n<p>Now, divide [latex]8\\div\\frac{1}{4}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}8\\div\\frac{1}{4}=\\frac{8}{1}\\cdot\\frac{4}{1}=32\\\\\\text{}\\\\1-32\\end{array}[\/latex]<\/p>\n<p>Subtract.<\/p>\n<p style=\"text-align: center;\">[latex]1\u201332=\u221231[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3\\cdot \\frac{1}{3}-8\\div \\frac{1}{4}=-31[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video you are shown how to use the order of operations to simplify an expression that contains multiplication, division, and subtraction with terms that contain fractions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-10\" title=\"Simplify an Expression in the Form:  a*1\/b-c\/(1\/d)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/yqp06obmcVc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Simplify expressions containing exponents<\/h3>\n<p>When you are evaluating expressions, you will sometimes see exponents used to represent repeated multiplication. Recall that an expression such as [latex]7^{2}[\/latex]\u00a0is <strong>exponential notation<\/strong> for [latex]7\\cdot7[\/latex]. (Exponential notation has two parts: the <strong>base<\/strong> and the <strong>exponent<\/strong> or the <strong>power<\/strong>. In [latex]7^{2}[\/latex], 7 is the base and 2 is the exponent; the exponent determines how many times the base is multiplied by itself.)<\/p>\n<p>Exponents are a way to represent repeated multiplication; the order of operations places it <i>before <\/i>any other multiplication, division, subtraction, and addition is performed.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 15<\/h3>\n<p>Simplify [latex]3^{2}\\cdot2^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q360237\">Show Solution<\/span><\/p>\n<div id=\"q360237\" class=\"hidden-answer\" style=\"display: none\">This problem has exponents and multiplication in it. According to the order of operations, simplifying\u00a0[latex]3^{2}[\/latex]\u00a0and [latex]2^{3}[\/latex]\u00a0comes before multiplication.<\/p>\n<p style=\"text-align: center;\">[latex]3^{2}\\cdot2^{3}[\/latex]<\/p>\n<p>[latex]{{3}^{2}}[\/latex] is [latex]3\\cdot3[\/latex], which equals 9.<\/p>\n<p style=\"text-align: center;\">[latex]9\\cdot {{2}^{3}}[\/latex]<\/p>\n<p>[latex]{{2}^{3}}[\/latex] is [latex]2\\cdot2\\cdot2[\/latex], which equals 8.<\/p>\n<p style=\"text-align: center;\">[latex]9\\cdot 8[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]9\\cdot 8=72[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]{{3}^{2}}\\cdot {{2}^{3}}=72[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the video that follows, an expression with exponents on its terms is simplified using the order of operations.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-11\" title=\"Simplify an Expression in the Form:  a^n*b^m\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/JjBBgV7G_Qw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Simplify expressions containing multiple grouping symbols<\/h3>\n<p>Grouping symbols such as parentheses ( ), brackets [ ], braces[latex]\\displaystyle \\left\\{ {} \\right\\}[\/latex], and fraction bars can be used to further control the order of the four arithmetic operations.\u00a0The rules of the order of operations require computation within grouping symbols to be completed first, even if you are adding or subtracting within the grouping symbols and you have multiplication outside the grouping symbols. After computing within the grouping symbols, divide or multiply from left to right and then subtract or add from left to right.\u00a0When there are grouping symbols within grouping symbols, calculate from the inside to the outside. That is, begin simplifying within the innermost grouping symbols first.<\/p>\n<p>Remember that parentheses can also be used to show multiplication. In the example that follows, both uses of parentheses\u2014as a way to represent a group, as well as a way to express multiplication\u2014are shown.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 16<\/h3>\n<p>Simplify [latex]\\left(3+4\\right)^{2}+\\left(8\\right)\\left(4\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q548490\">Show Solution<\/span><\/p>\n<div id=\"q548490\" class=\"hidden-answer\" style=\"display: none\">This problem has parentheses, exponents, multiplication, and addition in it. The first set of parentheses is a grouping symbol. The second set indicates multiplication.<\/p>\n<p>Grouping symbols are handled first. Add numbers in parentheses.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}(3+4)^{2}+(8)(4)\\\\(7)^{2}+(8)(4)\\end{array}[\/latex]<\/p>\n<p>Simplify\u00a0[latex]7^{2}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}7^{2}+(8)(4)\\\\49+(8)(4)\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}49+(8)(4)\\\\49+(32)\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Add.<\/p>\n<p style=\"text-align: center;\">[latex]49+32=81[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex](3+4)^{2}+(8)(4)=81[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example 17<\/h3>\n<p>Simplify \u00a0[latex]4\\cdot{\\frac{3[5+{(2 + 3)}^2]}{2}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q358226\">Show Solution<\/span><\/p>\n<div id=\"q358226\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are brackets and parentheses in this problem. Compute inside the innermost grouping symbols first.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}4\\cdot{\\frac{3[5+{(2 + 3)}^2]}{2}}\\\\\\text{ }\\\\=4\\cdot{\\frac{3[5+{(5)}^2]}{2}}\\end{array}[\/latex]<\/p>\n<p>Then apply the exponent<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}4\\cdot{\\frac{3[5+{(5)}^2]}{2}}\\\\\\text{}\\\\=4\\cdot{\\frac{3[5+25]}{2}}\\\\\\text{ }\\\\=4\\cdot{\\frac{3[30]}{2}}\\end{array}[\/latex]<\/p>\n<p>Then simplify the fraction<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}4\\cdot{\\frac{3[30]}{2}}\\\\\\text{}\\\\=4\\cdot{\\frac{90}{2}}\\\\\\text{ }\\\\=4\\cdot{45}\\\\\\text{ }\\\\=180\\end{array}[\/latex]<\/p>\n<h4 style=\"text-align: left;\">Answer<\/h4>\n<p>[latex]4\\cdot{\\frac{3[5+{(2 + 3)}^2]}{2}}=180[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you are shown how to use the order of operations to simplify an expression with grouping symbols, exponents, multiplication, and addition.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-12\" title=\"Simplify an Expression in the Form:  (a+b)^2+c*d\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/EMch2MKCVdA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>These problems are very similar to the examples given above. How are they different and what tools do you need to simplify them?<\/p>\n<p>a) Simplify\u00a0[latex]\\left(1.5+3.5\\right)\u20132\\left(0.5\\cdot6\\right)^{2}[\/latex].\u00a0This problem has parentheses, exponents, multiplication, subtraction, and addition in it, as well as decimals instead of integers.<\/p>\n<p>Use the box below to write down a few thoughts about how you would simplify this expression with decimals and grouping symbols.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q680970\">Show Solution<\/span><\/p>\n<div id=\"q680970\" class=\"hidden-answer\" style=\"display: none\">\nGrouping symbols are handled first. Add numbers in the first set of parentheses.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}(1.5+3.5)\u20132(0.5\\cdot6)^{2}\\\\5\u20132(0.5\\cdot6)^{2}\\end{array}[\/latex]<\/p>\n<p>Multiply numbers in the second set of parentheses.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}5\u20132(0.5\\cdot6)^{2}\\\\5\u20132(3)^{2}\\end{array}[\/latex]<\/p>\n<p>Evaluate exponents.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}5\u20132(3)^{2}\\\\5\u20132\\cdot9\\end{array}[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}5\u20132\\cdot9\\\\5\u201318\\end{array}[\/latex]<\/p>\n<p>Subtract.<\/p>\n<p style=\"text-align: center;\">[latex]5\u201318=\u221213[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex](1.5+3.5)\u20132(0.5\\cdot6)^{2}=\u221213[\/latex]<\/p>\n<\/div>\n<\/div>\n<p class=\"p1\">b) Simplify [latex]{{\\left( \\frac{1}{2} \\right)}^{2}}+{{\\left( \\frac{1}{4} \\right)}^{3}}\\cdot \\,32[\/latex].<\/p>\n<p>Use the box below to write down a few thoughts about how you would simplify this expression with fractions and grouping symbols.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q680972\">Show Solution<\/span><\/p>\n<div id=\"q680972\" class=\"hidden-answer\" style=\"display: none\">\nThis problem has exponents, multiplication, and addition in it, as well as fractions instead of integers.<\/p>\n<p>According to the order of operations, simplify the terms with the exponents first, then multiply, then add.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(\\frac{1}{2}\\right)^{2}+\\left(\\frac{1}{4}\\right)^{3}\\cdot32[\/latex]<\/p>\n<p>Evaluate: [latex]\\left(\\frac{1}{2}\\right)^{2}=\\frac{1}{2}\\cdot\\frac{1}{2}=\\frac{1}{4}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{4}+\\left(\\frac{1}{4}\\right)^{3}\\cdot32[\/latex]<\/p>\n<p>Evaluate: [latex]\\left(\\frac{1}{4}\\right)^{3}=\\frac{1}{4}\\cdot\\frac{1}{4}\\cdot\\frac{1}{4}=\\frac{1}{64}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{4}+\\frac{1}{64}\\cdot32[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{4}+\\frac{32}{64}[\/latex]<\/p>\n<p>Simplify. [latex]\\frac{32}{64}=\\frac{1}{2}[\/latex], so you can add [latex]\\frac{1}{4}+\\frac{1}{2}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{4}+\\frac{1}{2}=\\frac{3}{4}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]{{\\left( \\frac{1}{2} \\right)}^{2}}+{{\\left( \\frac{1}{4} \\right)}^{3}}\\cdot 32=\\frac{3}{4}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 18<\/h3>\n<p>Simplify [latex]\\frac{5-[3+(2\\cdot (-6))]}{{{3}^{2}}+2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q906386\">Show Solution<\/span><\/p>\n<div id=\"q906386\" class=\"hidden-answer\" style=\"display: none\">This problem has brackets, parentheses, fractions, exponents, multiplication, subtraction, and addition in it.<\/p>\n<p>Grouping symbols are handled first. The parentheses around the [latex]-6[\/latex] aren\u2019t a grouping symbol; they are simply making it clear that the negative sign belongs to the 6. Start with the innermost set of parentheses that are a grouping symbol. In this example, the innermost set of parentheses\u00a0would be in\u00a0the numerator of the fraction, [latex](2\\cdot(\u22126))[\/latex]. Begin working out from there. (The fraction line acts as a type of grouping symbol, too; you simplify the numerator and denominator independently, and then divide the numerator by the denominator at the end.)<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{5-\\left[3+\\left(2\\cdot\\left(-6\\right)\\right)\\right]}{3^{2}+2}\\\\\\\\\\frac{5-\\left[3+\\left(-12\\right)\\right]}{3^{2}+2}\\end{array}[\/latex]<\/p>\n<p>Add [latex]3[\/latex] and [latex]-12[\/latex], which are in brackets, to get [latex]-9[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{5-\\left[3+\\left(-12\\right)\\right]}{3^{2}+2}\\\\\\\\\\frac{5-\\left[-9\\right]}{3^{2}+2}\\end{array}[\/latex]<\/p>\n<p>Subtract [latex]5\u2013\\left[\u22129\\right]=5+9=14[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{5-\\left[-9\\right]}{3^{2}+2}\\\\\\\\\\frac{14}{3^{2}+2}\\end{array}[\/latex]<\/p>\n<p>The top of the fraction is all set, but the bottom (denominator) has remained untouched. Apply the order of operations to that as well. Begin by evaluating\u00a0[latex]3^{2}=9[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{14}{3^{2}+2}\\\\\\\\\\frac{14}{9+2}\\end{array}[\/latex]<\/p>\n<p>Now add. [latex]9+2=11[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{14}{9+2}\\\\\\\\\\frac{14}{11}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{5-\\left[3+\\left(2\\cdot\\left(-6\\right)\\right)\\right]}{3^{2}+2}=\\frac{14}{11}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The video that follows contains an example similar to the written one above. Note how the numerator and denominator of the fraction are simplified separately.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-13\" title=\"Simplify an Expression in Fraction Form\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/xIJLq54jM44?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h3>Simplify expressions containing absolute values<\/h3>\n<p>Absolute value expressions are one final method of grouping that you may see. Recall that the absolute value of a quantity is always positive or 0.<\/p>\n<p>When you see an absolute value expression included within a larger expression, treat the absolute value like a grouping symbol and evaluate the expression within the absolute value sign first. Then take the absolute value of that expression. The example below shows how this is done.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example 19<\/h3>\n<p>Simplify [latex]\\frac{3+\\left|2-6\\right|}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q572632\">Show Solution<\/span><\/p>\n<div id=\"q572632\" class=\"hidden-answer\" style=\"display: none\">This problem has absolute values, decimals, multiplication, subtraction, and addition in it.<\/p>\n<p>Grouping symbols, including absolute value, are handled first. Simplify the numerator, then the denominator.<\/p>\n<p>Evaluate [latex]\\left|2\u20136\\right|[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{3+\\left|2-6\\right|}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\\\\\\\\\frac{3+\\left|-4\\right|}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\end{array}[\/latex]<\/p>\n<p>Take the absolute value of [latex]\\left|\u22124\\right|[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{3+\\left|-4\\right|}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\\\\\\\\\frac{3+4}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\end{array}[\/latex]<\/p>\n<p>Add the numbers in the numerator.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{3+4}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\\\\\\\\\frac{7}{2\\left| 3\\cdot 1.5 \\right|-(-3)}\\end{array}[\/latex]<\/p>\n<p>Now that the numerator is simplified, turn to the denominator.<\/p>\n<p>Evaluate the absolute value expression first. [latex]3 \\cdot 1.5 = 4.5[\/latex], giving<\/p>\n<p style=\"text-align: center;\">\u00a0[latex]\\begin{array}{c}\\frac{7}{2\\left|{3\\cdot{1.5}}\\right|-(-3)}\\\\\\\\\\frac{7}{2\\left|{ 4.5}\\right|-(-3)}\\end{array}[\/latex]<\/p>\n<p>The expression \u201c[latex]2\\left|4.5\\right|[\/latex]\u201d reads \u201c2 times the absolute value of 4.5.\u201d Multiply 2 times 4.5.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{7}{2\\left|4.5\\right|-\\left(-3\\right)}\\\\\\\\\\frac{7}{9-\\left(-3\\right)}\\end{array}[\/latex]<\/p>\n<p>Subtract.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{7}{9-\\left(-3\\right)}\\\\\\\\\\frac{7}{12}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{3+\\left|2-6\\right|}{2\\left|3\\cdot1.5\\right|-3\\left(-3\\right)}=\\frac{7}{12}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The following video uses the order of operations to simplify an expression in fraction form that contains absolute value terms. Note how the absolute values are treated like parentheses and brackets when using the order of operations.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-14\" title=\"Simplify an Expression in Fraction Form with Absolute Values\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/6wmCQprxlnU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1830\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot Combo Meal. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 2: Subtracting Integers (Two Digit Integers). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ciuIKFCtWWU\">https:\/\/youtu.be\/ciuIKFCtWWU<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Find the Sum and Difference of Three Signed Fractions (Common Denom). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/P972VVbR98k\">https:\/\/youtu.be\/P972VVbR98k<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Adding Signed Decimals. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/3FHZQ5iKcpI\">https:\/\/youtu.be\/3FHZQ5iKcpI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 9: Real Numbers, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Multiplying Three or More Integers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/rx8F9SPd0HE\">https:\/\/youtu.be\/rx8F9SPd0HE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Multiplying Signed Fractions . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/yUdJ46pTblo\">https:\/\/youtu.be\/yUdJ46pTblo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 1: Dividing Signed Fractions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/OPHdadhDJoI\">https:\/\/youtu.be\/OPHdadhDJoI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 2: Combining Like Terms. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/b9-7eu29pNM\">https:\/\/youtu.be\/b9-7eu29pNM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify an Expression in the Form: a-b+c*d. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/yFO_0dlfy-w\">https:\/\/youtu.be\/yFO_0dlfy-w<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify an Expression in the Form: a*1\/b-c\/(1\/d). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/yqp06obmcVc\">https:\/\/youtu.be\/yqp06obmcVc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify an Expression in the Form: (a+b)^2+c*d. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/EMch2MKCVdA\">https:\/\/youtu.be\/EMch2MKCVdA<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify an Expression in Fraction Form. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/xIJLq54jM44\">https:\/\/youtu.be\/xIJLq54jM44<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify an Expression in Fraction Form with Absolute Values. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/6wmCQprxlnU\">https:\/\/youtu.be\/6wmCQprxlnU<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":20,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 2: Subtracting Integers (Two Digit Integers)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ciuIKFCtWWU\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Find the Sum and Difference of Three Signed Fractions (Common Denom)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen 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