{"id":16087,"date":"2019-10-01T17:13:42","date_gmt":"2019-10-01T17:13:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-use-properties-of-real-numbers\/"},"modified":"2020-09-11T00:34:55","modified_gmt":"2020-09-11T00:34:55","slug":"read-use-properties-of-real-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/read-use-properties-of-real-numbers\/","title":{"raw":"5.1.d - Use Properties of Real Numbers","rendered":"5.1.d &#8211; Use Properties of Real Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify expressions with real numbers that require all operations<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Order of Operations<\/h2>\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]Order of operations are a set of conventions used to provide a formulaic outline to follow when you are required to use several mathematical operations for one expression.\u00a0 The box below is a summary of the order of operations depicted in the graphic above.\r\n<div class=\"textbox shaded\">\r\n<h3>The Order of Operations<\/h3>\r\n<ul>\r\n \t<li>Perform all operations within grouping symbols first. Grouping symbols include parentheses ( ), brackets [ ], braces { }, and fraction bars.<\/li>\r\n \t<li>Evaluate exponents or square roots.<\/li>\r\n \t<li>Multiply or divide, from left to right.<\/li>\r\n \t<li>Add or subtract, from left to right.<\/li>\r\n<\/ul>\r\nThis order of operations is true for all real numbers.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/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<\/h3>\r\nSimplify [latex]3\\cdot\\dfrac{1}{3}\\normalsize -8\\div\\dfrac{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]\\left(3\\cdot\\dfrac{1}{3}\\normalsize\\right)-\\left(8\\div\\dfrac{1}{4}\\normalsize\\right)[\/latex]<\/p>\r\nMultiply [latex] 3\\cdot\\dfrac{1}{3}[\/latex] first.\r\n<p style=\"text-align: center\">[latex]\\left( 3\\cdot\\dfrac{1}{3}\\normalsize\\right)-\\left(8\\div\\dfrac{1}{4}\\normalsize\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\left(1\\right)-\\left(8\\div\\dfrac{1}{4}\\normalsize\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left\">Now, divide [latex]8\\div\\dfrac{1}{4}[\/latex].<\/p>\r\n<p style=\"text-align: center\">[latex]8\\div\\dfrac{1}{4}\\normalsize =\\dfrac{8}{1}\\normalsize\\cdot\\dfrac{4}{1}\\normalsize =32[\/latex]<\/p>\r\nSubtract.\r\n<p style=\"text-align: center\">[latex]\\left(1\\right)\u2013\\left(32\\right)=\u221231[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] 3\\cdot\\dfrac{1}{3}\\normalsize -8\\div\\dfrac{1}{4}\\normalsize =-31[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]149534[\/ohm_question]\r\n\r\n<\/div>\r\nIf the expression has exponents or square roots, they are to be performed a<i>fter <\/i>parentheses and other grouping symbols have been simplified and <i>before <\/i>any multiplication, division, subtraction, and addition that are outside the parentheses or other grouping symbols.\r\n<h2>Exponents<\/h2>\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], [latex]7[\/latex] is the base and [latex]2[\/latex] 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<\/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 [latex]9[\/latex].\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 [latex]8[\/latex].\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<h2>Grouping Symbols<\/h2>\r\nGrouping symbols such as parentheses ( ), brackets [ ], braces[latex] \\displaystyle \\left\\{ {} \\right\\}[\/latex], fraction bars, and roots 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<\/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<\/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\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]109960[\/ohm_question]\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>Try it<\/h3>\r\n[ohm_question]2320[\/ohm_question]\r\n\r\n<\/div>\r\nSquare roots are another grouping symbol.\u00a0 Operations inside of a square root need to be performed first. In the next example, we will simplify an expression that has a square root.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify\u00a0[latex]\\dfrac{\\sqrt{7+2}+2^2}{(8)(4)-11}[\/latex]\r\n\r\n[reveal-answer q=\"270259\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"270259\"]This problem has all the operations to consider with the order of operations.\r\n\r\nGrouping symbols are handled first, in this case the fraction bar. We will simplify the numerator (top) and denominator (bottom) separately.\r\nTo simplify the top:\r\n<p style=\"text-align: center\">[latex]\\sqrt{7+2}+2^2[\/latex]<\/p>\r\nAdd the numbers inside the square root (as they are essentially grouped by that symbol\/operator), and the term [latex]2^2[\/latex]\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\sqrt{(7+2)}+(2^2)\\\\\\\\=\\sqrt{9}+4\\\\\\\\=3+4=7\\end{array}[\/latex]<\/p>\r\nTo simplify the bottom:\r\n<p style=\"text-align: center\">[latex](8)(4)-11[\/latex]<\/p>\r\nMultiply\u00a0[latex]8[\/latex] and\u00a0[latex]4[\/latex] first, then subtract\u00a0[latex]11[\/latex].\r\n<p style=\"text-align: center\">[latex](8)(4)-11=[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\hspace{1cm}32-11=21[\/latex]<\/p>\r\n<p style=\"text-align: left\">Now put the fraction back together to see if any more simplifying needs to be done. The simplified numerator equaled [latex]7[\/latex], and the simplified denominator equaled [latex]21[\/latex].<\/p>\r\n<p style=\"text-align: center\">so [latex]\\dfrac{7}{21}[\/latex] , which can be reduced to [latex]\\dfrac{1}{3}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n<p style=\"text-align: center\">[latex]\\dfrac{\\sqrt{7+2}+2^2}{(8)(4)-11}=\\dfrac{1}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/9suc63qB96o\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\r\n&nbsp;\r\n<p class=\"p1\">b) Simplify [latex] {{\\left(\\dfrac{1}{2}\\normalsize\\right)}^{2}}+{{\\left(\\dfrac{1}{4}\\normalsize\\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(\\dfrac{1}{2}\\normalsize\\right)^{2}+\\left(\\dfrac{1}{4}\\normalsize\\right)^{3}\\cdot32[\/latex]<\/p>\r\nEvaluate: [latex]\\left(\\dfrac{1}{2}\\normalsize\\right)^{2}=\\dfrac{1}{2}\\normalsize\\cdot\\dfrac{1}{2}\\normalsize =\\dfrac{1}{4}[\/latex]\r\n<p style=\"text-align: center\">[latex]\\dfrac{1}{4}\\normalsize +\\left(\\dfrac{1}{4}\\normalsize\\right)^{3}\\cdot32[\/latex]<\/p>\r\nEvaluate: [latex]\\left(\\dfrac{1}{4}\\normalsize\\right)^{3}=\\frac{1}{4}\\normalsize\\cdot\\dfrac{1}{4}\\normalsize\\cdot\\dfrac{1}{4}\\normalsize=\\dfrac{1}{64}[\/latex]\r\n<p style=\"text-align: center\">[latex]\\dfrac{1}{4}\\normalsize +\\dfrac{1}{64}\\normalsize\\cdot32[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center\">[latex]\\dfrac{1}{4}\\normalsize +\\dfrac{32}{64}[\/latex]<\/p>\r\nSimplify. [latex]\\dfrac{32}{64}\\normalsize =\\dfrac{1}{2}[\/latex], so you can add [latex]\\dfrac{1}{4}\\normalsize +\\dfrac{1}{2}[\/latex].\r\n<p style=\"text-align: center\">[latex]\\dfrac{1}{4}\\normalsize +\\dfrac{1}{2}\\normalsize =\\dfrac{3}{4}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] {{\\left(\\dfrac{1}{2}\\normalsize\\right)}^{2}}+{{\\left(\\dfrac{1}{4} \\normalsize\\right)}^{3}}\\cdot 32=\\dfrac{3}{4}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]17577[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Combining Like Terms<\/h2>\r\nOne way we can simplify expressions is to combine like terms. <strong>Like terms<\/strong> are terms where the variables match exactly (exponents included). Examples of like terms would be [latex]5xy[\/latex] and [latex]-3xy[\/latex] or [latex]8a^2b[\/latex] and [latex]a^2b[\/latex] or [latex]-3[\/latex] and [latex]8[\/latex]. \u00a0If we have like terms we are\u00a0allowed to add (or subtract) the\u00a0numbers in front of the variables, then keep the variables the same. As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the following term like a negative term. The sign always stays with the term.\r\n\r\nThis is shown in the following examples:\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nCombine like terms: \u00a0[latex]5x-2y-8x+7y[\/latex]\r\n[reveal-answer q=\"730653\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"730653\"]\r\n\r\nThe like terms in this expression are:\r\n<p style=\"text-align: center\">[latex]5x[\/latex] and [latex]-8x[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]-2y[\/latex] and [latex]7y[\/latex]<\/p>\r\n<p style=\"text-align: left\">Note how we kept the sign in front of each term.<\/p>\r\n<p style=\"text-align: left\">Combine like terms:<\/p>\r\n<p style=\"text-align: center\">[latex]5x-8x = -3x[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]-2y+7y = 5y[\/latex]<\/p>\r\n<p style=\"text-align: left\">Note how signs become operations when you combine like terms.<\/p>\r\n<p style=\"text-align: left\">Simplified Expression:<\/p>\r\n<p style=\"text-align: center\">[latex]5x-2y-8x+7y=-3x+5y[\/latex]<\/p>\r\n<p style=\"text-align: center\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nIn the following video you will be shown how to combine like terms using the idea of the distributive property. \u00a0Note that this is a different method than is shown in the written examples on this page, but it obtains the same result.\r\n\r\nhttps:\/\/youtu.be\/JIleqbO8Tf0\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nCombine like terms: \u00a0[latex]x^2-3x+9-5x^2+3x-1[\/latex]\r\n\r\n[reveal-answer q=\"730650\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"730650\"]\r\n\r\nThe like terms in this expression are:\r\n<p style=\"text-align: center\">[latex]x^2[\/latex] and [latex]-5x^2[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]-3x[\/latex] and [latex]3x[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]9[\/latex] and [latex]-1[\/latex]<\/p>\r\n<p style=\"text-align: left\">Combine like terms:<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}x^2-5x^2 = -4x^2\\\\-3x+3x=0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\9-1=8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]-4x^2+8[\/latex]<\/p>\r\n<p style=\"text-align: center\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nIn the video that follows, you will be shown another example of\u00a0combining like terms. \u00a0Pay attention to why you are not able to combine all three terms in the example.\r\n\r\nhttps:\/\/youtu.be\/b9-7eu29pNM\r\n<h2><\/h2>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<h3>Distributive Property<\/h3>\r\nThe <strong>distributive property<\/strong> states that the product of a factor times a sum is the sum of the factor times each term in the sum.\r\n<div style=\"text-align: center\">[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/div>\r\nThis property combines both addition and multiplication (and is the only property to do so). Let us consider an example.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the distributive property to show that [latex]4\\cdot[12+(-7)]=20[\/latex]\r\n[reveal-answer q=\"907389\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"907389\"]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200212\/CNX_CAT_Figure_01_01_003.jpg\" alt=\"The number four is separated by a multiplication symbol from a bracketed expression reading: twelve plus negative seven. Arrows extend from the four pointing to the twelve and negative seven separately. This expression equals four times twelve plus four times negative seven. Under this line the expression reads forty eight plus negative twenty eight. Under this line the expression reads twenty as the answer. \" \/>\r\n\r\n<span style=\"font-size: 1rem;text-align: initial\">Note that [latex]4[\/latex] is outside the grouping symbols, so we distribute the \u00a0[latex]4[\/latex] by multiplying it by [latex]12[\/latex], multiplying it by [latex]\u20137[\/latex], and adding the products.<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nTo be more precise when describing this property, we say that multiplication distributes over addition.\r\n\r\nThe reverse is not true as we can see in this example.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}\\hfill 6+\\left(3\\cdot 5\\right)&amp; \\stackrel{?}{=}&amp; \\left(6+3\\right)\\cdot \\left(6+5\\right) \\\\ \\hfill 6+\\left(15\\right)&amp; \\stackrel{?}{=}&amp; \\left(9\\right)\\cdot \\left(11\\right)\\hfill \\\\ \\hfill 21&amp; \\ne &amp; \\text{ }99\\hfill \\end{array}[\/latex]<\/p>\r\nA special case of the distributive property occurs when a sum of terms is subtracted.\r\n<div style=\"text-align: center\">[latex]a-b=a+\\left(-b\\right)[\/latex]<\/div>\r\n<div style=\"text-align: center\">\r\n\r\nFor example, consider the difference [latex]12-\\left(5+3\\right)[\/latex]. We can rewrite the difference of the two terms [latex]12[\/latex] and [latex]\\left(5+3\\right)[\/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex]\\left(5+3\\right)[\/latex], we add the opposite.\r\n<div>[latex]12+\\left(-1\\right)\\cdot \\left(5+3\\right)[\/latex]<\/div>\r\nNow, distribute [latex]-1[\/latex] and simplify the result.\r\n<div>[latex]\\begin{array}{l}12-\\left(5+3\\right)=12+\\left(-1\\right)\\cdot\\left(5+3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3]\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+\\left(-8\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nRewrite the last example by changing the sign of each term and adding the results.\r\n[reveal-answer q=\"719333\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"719333\"]\r\n\r\n[latex]\\begin{array}{l}12-\\left(5+3\\right)=12+\\left(-5-3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+\\left(-8\\right) \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nThis seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms.\r\n<h3>Identity Properties<\/h3>\r\nThe <strong>identity property of addition<\/strong> states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.\r\n<div style=\"text-align: center\">[latex]a+0=a[\/latex]<\/div>\r\nThe <strong>identity property of multiplication<\/strong> states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.\r\n<div style=\"text-align: center\">[latex]a\\cdot 1=a[\/latex]<\/div>\r\n<div style=\"text-align: center\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p style=\"text-align: left\">Show that the identity property of addition and multiplication are true for [latex]-6 \\text{ and }23[\/latex].<\/p>\r\n<p style=\"text-align: left\">[reveal-answer q=\"587790\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"587790\"]<\/p>\r\n<p style=\"text-align: left\">[latex]\\left(-6\\right)+0=-6[\/latex]<\/p>\r\n<p style=\"text-align: left\">[latex]23+0=23[\/latex]<\/p>\r\n<p style=\"text-align: left\">[latex]-6\\cdot1=-6[\/latex]<\/p>\r\n<p style=\"text-align: left\">[latex]23\\cdot 1=23[\/latex]<\/p>\r\n<p style=\"text-align: left\">There are no exceptions for these properties; they work for every real number, including [latex]0[\/latex] and [latex]1[\/latex].<\/p>\r\n<p style=\"text-align: left\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<span style=\"color: #6c64ad;font-size: 1em;font-weight: 600\">Inverse Properties<\/span>\r\n\r\n<\/div>\r\nThe <strong>inverse property of addition<\/strong> states that, for every real number <em>a<\/em>, there is a unique number, called the additive inverse (or opposite), denoted\u00a0<em>a<\/em>, that, when added to the original number, results in the additive identity, [latex]0[\/latex].\r\n<div style=\"text-align: center\">[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\r\nFor example, if [latex]a=-8[\/latex], the additive inverse is [latex]8[\/latex], since [latex]\\left(-8\\right)+8=0[\/latex].\r\n\r\nThe <strong>inverse property of multiplication<\/strong> holds for all real numbers except [latex]0[\/latex] because the reciprocal of [latex]0[\/latex] is not defined. The property states that, for every real number <em>a<\/em>, there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\\dfrac{1}{a}[\/latex], that, when multiplied by the original number, results in the multiplicative identity, [latex]1[\/latex].\r\n<div style=\"text-align: center\">[latex]a\\cdot\\dfrac{1}{a}\\normalsize =1[\/latex]<\/div>\r\n<div style=\"text-align: center\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p style=\"text-align: left\">1) Define the additive inverse of\u00a0[latex]a=-8[\/latex], and use it to illustrate the inverse property of addition.<\/p>\r\n<p style=\"text-align: left\">2) Write the reciprocal of\u00a0[latex]a=-\\dfrac{2}{3}[\/latex], and use it to illustrate the inverse property of multiplication.<\/p>\r\n<p style=\"text-align: left\">[reveal-answer q=\"468875\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"468875\"]<\/p>\r\n<p style=\"text-align: left\">1) The additive inverse is [latex]8[\/latex], and\u00a0[latex]\\left(-8\\right)+8=0[\/latex]<\/p>\r\n<p style=\"text-align: left\">2) The reciprocal is [latex]-\\dfrac{3}{2}[\/latex]\u00a0and\u00a0[latex]\\left(-\\dfrac{2}{3}\\normalsize\\right)\\cdot \\left(-\\dfrac{3}{2}\\normalsize\\right)=1[\/latex]<\/p>\r\n<p style=\"text-align: left\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3 style=\"text-align: left\">A General Note: Properties of Real Numbers<\/h3>\r\nThe following properties hold for real numbers <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>.\r\n<table style=\"width: 70%\" summary=\"A table with six rows and three columns. The first entry of the first row is blank while the remaining columns read: Addition and Multiplication. The first entry of the second row reads: Commutative Property. The second column entry reads a plus b equals b plus a. The third column entry reads a times b equals b times a. The first entry of the third row reads Associative Property. The second column entry reads: a plus the quantity b plus c in parenthesis equals the quantity a plus b in parenthesis plus c. The third column entry reads: a times the quantity b times c in parenthesis equals the quantity a times b in parenthesis times c. The first entry of the fourth row reads: Distributive Property. The second and third column are combined on this row and read: a times the quantity b plus c in parenthesis equals a times b plus a times c. The first entry in the fifth row reads: Identity Property. The second column entry reads: There exists a unique real number called the additive identity, 0, such that for any real number a, a + 0 = a. The third column entry reads: There exists a unique real number called the multiplicative inverse, 1, such that for any real number a, a times 1 equals a. The first entry in the sixth row reads: Inverse Property. The second column entry reads: Every real number a has an additive inverse, or opposite, denoted negative a such that, a plus negative a equals zero. The third column entry reads: Every nonzero real\">\r\n<tbody>\r\n<tr>\r\n<th style=\"text-align: center\"><\/th>\r\n<th style=\"text-align: center\"><strong>Addition<\/strong><\/th>\r\n<th style=\"text-align: center\"><strong>Multiplication<\/strong><\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<tbody>\r\n<tr>\r\n<td><strong>Commutative Property<\/strong><\/td>\r\n<td>[latex]a+b=b+a[\/latex]<\/td>\r\n<td>[latex]a\\cdot b=b\\cdot a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Associative Property<\/strong><\/td>\r\n<td>[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/td>\r\n<td>[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Distributive Property<\/strong><\/td>\r\n<td>[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Identity Property<\/strong><\/td>\r\n<td>There exists a unique real number called the additive identity, 0, such that, for any real number <em>a<\/em>\r\n<div style=\"text-align: center\">[latex]a+0=a[\/latex]<\/div><\/td>\r\n<td>There exists a unique real number called the multiplicative identity, 1, such that, for any real number <em>a<\/em>\r\n<div style=\"text-align: center\">[latex]a\\cdot 1=a[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Inverse Property<\/strong><\/td>\r\n<td>Every real number a has an additive inverse, or opposite, denoted [latex]\u2013a[\/latex], such that\r\n<div style=\"text-align: center\">[latex]a+\\left(-a\\right)=0[\/latex]<\/div><\/td>\r\n<td>Every nonzero real number <em>a<\/em> has a multiplicative inverse, or reciprocal, denoted [latex]\\dfrac{1}{a}[\/latex], such that\r\n<div style=\"text-align: center\">[latex]a\\cdot \\left(\\dfrac{1}{a}\\normalsize\\right)=1[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the properties of real numbers to rewrite and simplify each expression. State which properties apply.\r\n<ol style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]3\\left(6+4\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(5+8\\right)+\\left(-8\\right)[\/latex]<\/li>\r\n \t<li>[latex]6-\\left(15+9\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{4}{7}\\normalsize\\cdot \\left(\\dfrac{2}{3}\\normalsize\\cdot\\dfrac{7}{4}\\normalsize\\right)[\/latex]<\/li>\r\n \t<li>[latex]100\\cdot \\left[0.75+\\left(-2.38\\right)\\right][\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"823624\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"823624\"]\r\n<ol style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\3\\cdot\\left(6+4\\right)=3\\cdot6+3\\cdot4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Distributive property} \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=18+12\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify} \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=30\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify}\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\\\left(5+8\\right)+\\left(-8\\right)=5+\\left[8+\\left(-8\\right)\\right]\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Associative property of addition} \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=5+0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Inverse property of addition} \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Identity property of addition}\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\6-\\left(15+9\\right) \\hfill&amp; =6+[\\left(-15\\right)+\\left(-9\\right)] \\hfill&amp; \\text{Distributive property} \\\\ \\hfill&amp; =6+\\left(-24\\right) \\hfill&amp; \\text{Simplify} \\\\ \\hfill&amp; =-18 \\hfill&amp; \\text{Simplify}\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\\\\\\\\\\\frac{4}{7}\\cdot\\left(\\frac{2}{3}\\cdot\\frac{7}{4}\\right) \\hfill&amp; =\\frac{4}{7} \\cdot\\left(\\frac{7}{4}\\cdot\\frac{2}{3}\\right) \\hfill&amp; \\text{Commutative property of multiplication} \\\\ \\hfill&amp; =\\left(\\frac{4}{7}\\cdot\\frac{7}{4}\\right)\\cdot\\frac{2}{3}\\hfill&amp; \\text{Associative property of multiplication} \\\\ \\hfill&amp; =1\\cdot\\frac{2}{3} \\hfill&amp; \\text{Inverse property of multiplication} \\\\ \\hfill&amp; =\\frac{2}{3} \\hfill&amp; \\text{Identity property of multiplication}\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\100\\cdot[0.75+\\left(-2.38\\right)] \\hfill&amp; =100\\cdot0.75+100\\cdot\\left(-2.38\\right)\\hfill&amp; \\text{Distributive property} \\\\ \\hfill&amp; =75+\\left(-238\\right) \\hfill&amp; \\text{Simplify} \\\\ \\hfill&amp; =-163 \\hfill&amp; \\text{Simplify}\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/8SFm8Os_4C8","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify expressions with real numbers that require all operations<\/li>\n<\/ul>\n<\/div>\n<h2>Order of Operations<\/h2>\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<p>Order of operations are a set of conventions used to provide a formulaic outline to follow when you are required to use several mathematical operations for one expression.\u00a0 The box below is a summary of the order of operations depicted in the graphic above.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Order of Operations<\/h3>\n<ul>\n<li>Perform all operations within grouping symbols first. Grouping symbols include parentheses ( ), brackets [ ], braces { }, and fraction bars.<\/li>\n<li>Evaluate exponents or square roots.<\/li>\n<li>Multiply or divide, from left to right.<\/li>\n<li>Add or subtract, from left to right.<\/li>\n<\/ul>\n<p>This order of operations is true for all real numbers.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/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-1\" 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<\/h3>\n<p>Simplify [latex]3\\cdot\\dfrac{1}{3}\\normalsize -8\\div\\dfrac{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]\\left(3\\cdot\\dfrac{1}{3}\\normalsize\\right)-\\left(8\\div\\dfrac{1}{4}\\normalsize\\right)[\/latex]<\/p>\n<p>Multiply [latex]3\\cdot\\dfrac{1}{3}[\/latex] first.<\/p>\n<p style=\"text-align: center\">[latex]\\left( 3\\cdot\\dfrac{1}{3}\\normalsize\\right)-\\left(8\\div\\dfrac{1}{4}\\normalsize\\right)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\left(1\\right)-\\left(8\\div\\dfrac{1}{4}\\normalsize\\right)[\/latex]<\/p>\n<p style=\"text-align: left\">Now, divide [latex]8\\div\\dfrac{1}{4}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]8\\div\\dfrac{1}{4}\\normalsize =\\dfrac{8}{1}\\normalsize\\cdot\\dfrac{4}{1}\\normalsize =32[\/latex]<\/p>\n<p>Subtract.<\/p>\n<p style=\"text-align: center\">[latex]\\left(1\\right)\u2013\\left(32\\right)=\u221231[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3\\cdot\\dfrac{1}{3}\\normalsize -8\\div\\dfrac{1}{4}\\normalsize =-31[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm149534\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=149534&theme=oea&iframe_resize_id=ohm149534&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>If the expression has exponents or square roots, they are to be performed a<i>fter <\/i>parentheses and other grouping symbols have been simplified and <i>before <\/i>any multiplication, division, subtraction, and addition that are outside the parentheses or other grouping symbols.<\/p>\n<h2>Exponents<\/h2>\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], [latex]7[\/latex] is the base and [latex]2[\/latex] 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<\/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 [latex]9[\/latex].<\/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 [latex]8[\/latex].<\/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-2\" 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<h2>Grouping Symbols<\/h2>\n<p>Grouping symbols such as parentheses ( ), brackets [ ], braces[latex]\\displaystyle \\left\\{ {} \\right\\}[\/latex], fraction bars, and roots 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<\/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<\/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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm109960\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109960&theme=oea&iframe_resize_id=ohm109960&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-3\" 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>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm2320\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2320&theme=oea&iframe_resize_id=ohm2320&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Square roots are another grouping symbol.\u00a0 Operations inside of a square root need to be performed first. In the next example, we will simplify an expression that has a square root.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify\u00a0[latex]\\dfrac{\\sqrt{7+2}+2^2}{(8)(4)-11}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q270259\">Show Solution<\/span><\/p>\n<div id=\"q270259\" class=\"hidden-answer\" style=\"display: none\">This problem has all the operations to consider with the order of operations.<\/p>\n<p>Grouping symbols are handled first, in this case the fraction bar. We will simplify the numerator (top) and denominator (bottom) separately.<br \/>\nTo simplify the top:<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{7+2}+2^2[\/latex]<\/p>\n<p>Add the numbers inside the square root (as they are essentially grouped by that symbol\/operator), and the term [latex]2^2[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\sqrt{(7+2)}+(2^2)\\\\\\\\=\\sqrt{9}+4\\\\\\\\=3+4=7\\end{array}[\/latex]<\/p>\n<p>To simplify the bottom:<\/p>\n<p style=\"text-align: center\">[latex](8)(4)-11[\/latex]<\/p>\n<p>Multiply\u00a0[latex]8[\/latex] and\u00a0[latex]4[\/latex] first, then subtract\u00a0[latex]11[\/latex].<\/p>\n<p style=\"text-align: center\">[latex](8)(4)-11=[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\hspace{1cm}32-11=21[\/latex]<\/p>\n<p style=\"text-align: left\">Now put the fraction back together to see if any more simplifying needs to be done. The simplified numerator equaled [latex]7[\/latex], and the simplified denominator equaled [latex]21[\/latex].<\/p>\n<p style=\"text-align: center\">so [latex]\\dfrac{7}{21}[\/latex] , which can be reduced to [latex]\\dfrac{1}{3}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p style=\"text-align: center\">[latex]\\dfrac{\\sqrt{7+2}+2^2}{(8)(4)-11}=\\dfrac{1}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Order of Operations with a Fraction Containing a Square Root\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/9suc63qB96o?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>&nbsp;<\/p>\n<p class=\"p1\">b) Simplify [latex]{{\\left(\\dfrac{1}{2}\\normalsize\\right)}^{2}}+{{\\left(\\dfrac{1}{4}\\normalsize\\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(\\dfrac{1}{2}\\normalsize\\right)^{2}+\\left(\\dfrac{1}{4}\\normalsize\\right)^{3}\\cdot32[\/latex]<\/p>\n<p>Evaluate: [latex]\\left(\\dfrac{1}{2}\\normalsize\\right)^{2}=\\dfrac{1}{2}\\normalsize\\cdot\\dfrac{1}{2}\\normalsize =\\dfrac{1}{4}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{1}{4}\\normalsize +\\left(\\dfrac{1}{4}\\normalsize\\right)^{3}\\cdot32[\/latex]<\/p>\n<p>Evaluate: [latex]\\left(\\dfrac{1}{4}\\normalsize\\right)^{3}=\\frac{1}{4}\\normalsize\\cdot\\dfrac{1}{4}\\normalsize\\cdot\\dfrac{1}{4}\\normalsize=\\dfrac{1}{64}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{1}{4}\\normalsize +\\dfrac{1}{64}\\normalsize\\cdot32[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{1}{4}\\normalsize +\\dfrac{32}{64}[\/latex]<\/p>\n<p>Simplify. [latex]\\dfrac{32}{64}\\normalsize =\\dfrac{1}{2}[\/latex], so you can add [latex]\\dfrac{1}{4}\\normalsize +\\dfrac{1}{2}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{1}{4}\\normalsize +\\dfrac{1}{2}\\normalsize =\\dfrac{3}{4}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]{{\\left(\\dfrac{1}{2}\\normalsize\\right)}^{2}}+{{\\left(\\dfrac{1}{4} \\normalsize\\right)}^{3}}\\cdot 32=\\dfrac{3}{4}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm17577\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=17577&theme=oea&iframe_resize_id=ohm17577&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Combining Like Terms<\/h2>\n<p>One way we can simplify expressions is to combine like terms. <strong>Like terms<\/strong> are terms where the variables match exactly (exponents included). Examples of like terms would be [latex]5xy[\/latex] and [latex]-3xy[\/latex] or [latex]8a^2b[\/latex] and [latex]a^2b[\/latex] or [latex]-3[\/latex] and [latex]8[\/latex]. \u00a0If we have like terms we are\u00a0allowed to add (or subtract) the\u00a0numbers in front of the variables, then keep the variables the same. As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the following term like a negative term. The sign always stays with the term.<\/p>\n<p>This is shown in the following examples:<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Combine like terms: \u00a0[latex]5x-2y-8x+7y[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q730653\">Show Solution<\/span><\/p>\n<div id=\"q730653\" class=\"hidden-answer\" style=\"display: none\">\n<p>The like terms in this expression are:<\/p>\n<p style=\"text-align: center\">[latex]5x[\/latex] and [latex]-8x[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]-2y[\/latex] and [latex]7y[\/latex]<\/p>\n<p style=\"text-align: left\">Note how we kept the sign in front of each term.<\/p>\n<p style=\"text-align: left\">Combine like terms:<\/p>\n<p style=\"text-align: center\">[latex]5x-8x = -3x[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]-2y+7y = 5y[\/latex]<\/p>\n<p style=\"text-align: left\">Note how signs become operations when you combine like terms.<\/p>\n<p style=\"text-align: left\">Simplified Expression:<\/p>\n<p style=\"text-align: center\">[latex]5x-2y-8x+7y=-3x+5y[\/latex]<\/p>\n<p style=\"text-align: center\"><\/div>\n<\/div>\n<\/div>\n<p>In the following video you will be shown how to combine like terms using the idea of the distributive property. \u00a0Note that this is a different method than is shown in the written examples on this page, but it obtains the same result.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex 1:  Combining Like Terms\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/JIleqbO8Tf0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Combine like terms: \u00a0[latex]x^2-3x+9-5x^2+3x-1[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q730650\">Show Solution<\/span><\/p>\n<div id=\"q730650\" class=\"hidden-answer\" style=\"display: none\">\n<p>The like terms in this expression are:<\/p>\n<p style=\"text-align: center\">[latex]x^2[\/latex] and [latex]-5x^2[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]-3x[\/latex] and [latex]3x[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]9[\/latex] and [latex]-1[\/latex]<\/p>\n<p style=\"text-align: left\">Combine like terms:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}x^2-5x^2 = -4x^2\\\\-3x+3x=0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\9-1=8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]-4x^2+8[\/latex]<\/p>\n<p style=\"text-align: center\"><\/div>\n<\/div>\n<\/div>\n<p>In the video that follows, you will be shown another example of\u00a0combining like terms. \u00a0Pay attention to why you are not able to combine all three terms in the example.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Ex 2:  Combining Like Terms\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/b9-7eu29pNM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><\/h2>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<h3>Distributive Property<\/h3>\n<p>The <strong>distributive property<\/strong> states that the product of a factor times a sum is the sum of the factor times each term in the sum.<\/p>\n<div style=\"text-align: center\">[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/div>\n<p>This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the distributive property to show that [latex]4\\cdot[12+(-7)]=20[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q907389\">Show Solution<\/span><\/p>\n<div id=\"q907389\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200212\/CNX_CAT_Figure_01_01_003.jpg\" alt=\"The number four is separated by a multiplication symbol from a bracketed expression reading: twelve plus negative seven. Arrows extend from the four pointing to the twelve and negative seven separately. This expression equals four times twelve plus four times negative seven. Under this line the expression reads forty eight plus negative twenty eight. Under this line the expression reads twenty as the answer.\" \/><\/p>\n<p><span style=\"font-size: 1rem;text-align: initial\">Note that [latex]4[\/latex] is outside the grouping symbols, so we distribute the \u00a0[latex]4[\/latex] by multiplying it by [latex]12[\/latex], multiplying it by [latex]\u20137[\/latex], and adding the products.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To be more precise when describing this property, we say that multiplication distributes over addition.<\/p>\n<p>The reverse is not true as we can see in this example.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}\\hfill 6+\\left(3\\cdot 5\\right)& \\stackrel{?}{=}& \\left(6+3\\right)\\cdot \\left(6+5\\right) \\\\ \\hfill 6+\\left(15\\right)& \\stackrel{?}{=}& \\left(9\\right)\\cdot \\left(11\\right)\\hfill \\\\ \\hfill 21& \\ne & \\text{ }99\\hfill \\end{array}[\/latex]<\/p>\n<p>A special case of the distributive property occurs when a sum of terms is subtracted.<\/p>\n<div style=\"text-align: center\">[latex]a-b=a+\\left(-b\\right)[\/latex]<\/div>\n<div style=\"text-align: center\">\n<p>For example, consider the difference [latex]12-\\left(5+3\\right)[\/latex]. We can rewrite the difference of the two terms [latex]12[\/latex] and [latex]\\left(5+3\\right)[\/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex]\\left(5+3\\right)[\/latex], we add the opposite.<\/p>\n<div>[latex]12+\\left(-1\\right)\\cdot \\left(5+3\\right)[\/latex]<\/div>\n<p>Now, distribute [latex]-1[\/latex] and simplify the result.<\/p>\n<div>[latex]\\begin{array}{l}12-\\left(5+3\\right)=12+\\left(-1\\right)\\cdot\\left(5+3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3]\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+\\left(-8\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Rewrite the last example by changing the sign of each term and adding the results.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q719333\">Show Solution<\/span><\/p>\n<div id=\"q719333\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{array}{l}12-\\left(5+3\\right)=12+\\left(-5-3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+\\left(-8\\right) \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms.<\/p>\n<h3>Identity Properties<\/h3>\n<p>The <strong>identity property of addition<\/strong> states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.<\/p>\n<div style=\"text-align: center\">[latex]a+0=a[\/latex]<\/div>\n<p>The <strong>identity property of multiplication<\/strong> states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.<\/p>\n<div style=\"text-align: center\">[latex]a\\cdot 1=a[\/latex]<\/div>\n<div style=\"text-align: center\">\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p style=\"text-align: left\">Show that the identity property of addition and multiplication are true for [latex]-6 \\text{ and }23[\/latex].<\/p>\n<p style=\"text-align: left\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q587790\">Show Solution<\/span><\/p>\n<div id=\"q587790\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left\">[latex]\\left(-6\\right)+0=-6[\/latex]<\/p>\n<p style=\"text-align: left\">[latex]23+0=23[\/latex]<\/p>\n<p style=\"text-align: left\">[latex]-6\\cdot1=-6[\/latex]<\/p>\n<p style=\"text-align: left\">[latex]23\\cdot 1=23[\/latex]<\/p>\n<p style=\"text-align: left\">There are no exceptions for these properties; they work for every real number, including [latex]0[\/latex] and [latex]1[\/latex].<\/p>\n<p style=\"text-align: left\"><\/div>\n<\/div>\n<\/div>\n<p><span style=\"color: #6c64ad;font-size: 1em;font-weight: 600\">Inverse Properties<\/span><\/p>\n<\/div>\n<p>The <strong>inverse property of addition<\/strong> states that, for every real number <em>a<\/em>, there is a unique number, called the additive inverse (or opposite), denoted\u00a0<em>a<\/em>, that, when added to the original number, results in the additive identity, [latex]0[\/latex].<\/p>\n<div style=\"text-align: center\">[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\n<p>For example, if [latex]a=-8[\/latex], the additive inverse is [latex]8[\/latex], since [latex]\\left(-8\\right)+8=0[\/latex].<\/p>\n<p>The <strong>inverse property of multiplication<\/strong> holds for all real numbers except [latex]0[\/latex] because the reciprocal of [latex]0[\/latex] is not defined. The property states that, for every real number <em>a<\/em>, there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\\dfrac{1}{a}[\/latex], that, when multiplied by the original number, results in the multiplicative identity, [latex]1[\/latex].<\/p>\n<div style=\"text-align: center\">[latex]a\\cdot\\dfrac{1}{a}\\normalsize =1[\/latex]<\/div>\n<div style=\"text-align: center\">\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p style=\"text-align: left\">1) Define the additive inverse of\u00a0[latex]a=-8[\/latex], and use it to illustrate the inverse property of addition.<\/p>\n<p style=\"text-align: left\">2) Write the reciprocal of\u00a0[latex]a=-\\dfrac{2}{3}[\/latex], and use it to illustrate the inverse property of multiplication.<\/p>\n<p style=\"text-align: left\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q468875\">Show Solution<\/span><\/p>\n<div id=\"q468875\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left\">1) The additive inverse is [latex]8[\/latex], and\u00a0[latex]\\left(-8\\right)+8=0[\/latex]<\/p>\n<p style=\"text-align: left\">2) The reciprocal is [latex]-\\dfrac{3}{2}[\/latex]\u00a0and\u00a0[latex]\\left(-\\dfrac{2}{3}\\normalsize\\right)\\cdot \\left(-\\dfrac{3}{2}\\normalsize\\right)=1[\/latex]<\/p>\n<p style=\"text-align: left\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3 style=\"text-align: left\">A General Note: Properties of Real Numbers<\/h3>\n<p>The following properties hold for real numbers <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>.<\/p>\n<table style=\"width: 70%\" summary=\"A table with six rows and three columns. The first entry of the first row is blank while the remaining columns read: Addition and Multiplication. The first entry of the second row reads: Commutative Property. The second column entry reads a plus b equals b plus a. The third column entry reads a times b equals b times a. The first entry of the third row reads Associative Property. The second column entry reads: a plus the quantity b plus c in parenthesis equals the quantity a plus b in parenthesis plus c. The third column entry reads: a times the quantity b times c in parenthesis equals the quantity a times b in parenthesis times c. The first entry of the fourth row reads: Distributive Property. The second and third column are combined on this row and read: a times the quantity b plus c in parenthesis equals a times b plus a times c. The first entry in the fifth row reads: Identity Property. The second column entry reads: There exists a unique real number called the additive identity, 0, such that for any real number a, a + 0 = a. The third column entry reads: There exists a unique real number called the multiplicative inverse, 1, such that for any real number a, a times 1 equals a. The first entry in the sixth row reads: Inverse Property. The second column entry reads: Every real number a has an additive inverse, or opposite, denoted negative a such that, a plus negative a equals zero. The third column entry reads: Every nonzero real\">\n<tbody>\n<tr>\n<th style=\"text-align: center\"><\/th>\n<th style=\"text-align: center\"><strong>Addition<\/strong><\/th>\n<th style=\"text-align: center\"><strong>Multiplication<\/strong><\/th>\n<\/tr>\n<\/tbody>\n<tbody>\n<tr>\n<td><strong>Commutative Property<\/strong><\/td>\n<td>[latex]a+b=b+a[\/latex]<\/td>\n<td>[latex]a\\cdot b=b\\cdot a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Associative Property<\/strong><\/td>\n<td>[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/td>\n<td>[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Distributive Property<\/strong><\/td>\n<td>[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><strong>Identity Property<\/strong><\/td>\n<td>There exists a unique real number called the additive identity, 0, such that, for any real number <em>a<\/em><\/p>\n<div style=\"text-align: center\">[latex]a+0=a[\/latex]<\/div>\n<\/td>\n<td>There exists a unique real number called the multiplicative identity, 1, such that, for any real number <em>a<\/em><\/p>\n<div style=\"text-align: center\">[latex]a\\cdot 1=a[\/latex]<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td><strong>Inverse Property<\/strong><\/td>\n<td>Every real number a has an additive inverse, or opposite, denoted [latex]\u2013a[\/latex], such that<\/p>\n<div style=\"text-align: center\">[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\n<\/td>\n<td>Every nonzero real number <em>a<\/em> has a multiplicative inverse, or reciprocal, denoted [latex]\\dfrac{1}{a}[\/latex], such that<\/p>\n<div style=\"text-align: center\">[latex]a\\cdot \\left(\\dfrac{1}{a}\\normalsize\\right)=1[\/latex]<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.<\/p>\n<ol style=\"list-style-type: lower-alpha\">\n<li>[latex]3\\left(6+4\\right)[\/latex]<\/li>\n<li>[latex]\\left(5+8\\right)+\\left(-8\\right)[\/latex]<\/li>\n<li>[latex]6-\\left(15+9\\right)[\/latex]<\/li>\n<li>[latex]\\dfrac{4}{7}\\normalsize\\cdot \\left(\\dfrac{2}{3}\\normalsize\\cdot\\dfrac{7}{4}\\normalsize\\right)[\/latex]<\/li>\n<li>[latex]100\\cdot \\left[0.75+\\left(-2.38\\right)\\right][\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q823624\">Show Solution<\/span><\/p>\n<div id=\"q823624\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha\">\n<li>[latex]\\begin{array}{l}\\\\\\\\3\\cdot\\left(6+4\\right)=3\\cdot6+3\\cdot4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Distributive property} \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=18+12\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify} \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=30\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify}\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{l}\\\\\\\\\\left(5+8\\right)+\\left(-8\\right)=5+\\left[8+\\left(-8\\right)\\right]\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Associative property of addition} \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=5+0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Inverse property of addition} \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Identity property of addition}\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{l}\\\\\\\\6-\\left(15+9\\right) \\hfill& =6+[\\left(-15\\right)+\\left(-9\\right)] \\hfill& \\text{Distributive property} \\\\ \\hfill& =6+\\left(-24\\right) \\hfill& \\text{Simplify} \\\\ \\hfill& =-18 \\hfill& \\text{Simplify}\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{l}\\\\\\\\\\\\\\\\\\frac{4}{7}\\cdot\\left(\\frac{2}{3}\\cdot\\frac{7}{4}\\right) \\hfill& =\\frac{4}{7} \\cdot\\left(\\frac{7}{4}\\cdot\\frac{2}{3}\\right) \\hfill& \\text{Commutative property of multiplication} \\\\ \\hfill& =\\left(\\frac{4}{7}\\cdot\\frac{7}{4}\\right)\\cdot\\frac{2}{3}\\hfill& \\text{Associative property of multiplication} \\\\ \\hfill& =1\\cdot\\frac{2}{3} \\hfill& \\text{Inverse property of multiplication} \\\\ \\hfill& =\\frac{2}{3} \\hfill& \\text{Identity property of multiplication}\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{l}\\\\\\\\100\\cdot[0.75+\\left(-2.38\\right)] \\hfill& =100\\cdot0.75+100\\cdot\\left(-2.38\\right)\\hfill& \\text{Distributive property} \\\\ \\hfill& =75+\\left(-238\\right) \\hfill& \\text{Simplify} \\\\ \\hfill& =-163 \\hfill& \\text{Simplify}\\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Properties of Real Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8SFm8Os_4C8?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-16087\">\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>Properties of Real Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/8SFm8Os_4C8\">https:\/\/youtu.be\/8SFm8Os_4C8<\/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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra: Using Properties of Real Numbers. <strong>Located at<\/strong>: <a 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