{"id":36,"date":"2023-06-21T13:22:26","date_gmt":"2023-06-21T13:22:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/order-of-operations-and-combining-like-terms\/"},"modified":"2023-08-07T01:29:34","modified_gmt":"2023-08-07T01:29:34","slug":"order-of-operations-and-combining-like-terms","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/order-of-operations-and-combining-like-terms\/","title":{"raw":"RP1.3   Combining Like Terms and Order of Operations","rendered":"RP1.3   Combining Like Terms and Order of Operations"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Recognize and combine like terms in an expression<\/li>\r\n \t<li>Use the order of operations to simplify expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2><\/h2>\r\nSome important terminology before we begin:\r\n<ul>\r\n \t<li><strong>operations\/operators:<\/strong>\u00a0In mathematics we call things like multiplication, division, addition, and subtraction operations. \u00a0They are the verbs of the math world, doing work on numbers and variables. The symbols used to denote operations are called operators, such as [latex]+{, }-{, }\\times{, }\\div[\/latex]. As you learn more math, you will learn more operators.<\/li>\r\n \t<li><strong>term:\u00a0<\/strong>Examples of terms would be [latex]2x[\/latex] and [latex]-\\Large\\frac{3}{2}[\/latex] or [latex]a^3[\/latex]. Even lone integers can be a term, like [latex]0[\/latex].<\/li>\r\n \t<li><strong>expression:\u00a0<\/strong>A\u00a0mathematical expression is one that connects terms with mathematical operators.\u00a0For example \u00a0[latex]\\Large\\frac{1}{2}\\normalsize +\\left(2^2\\right)- 9\\div\\Large\\frac{6}{7}[\/latex] is an expression.<\/li>\r\n<\/ul>\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\nSimplified Expression:\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>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.\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\\Large\\frac{1}{3}\\normalsize -8\\div\\Large\\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]\\left(3\\cdot\\Large\\frac{1}{3}\\normalsize\\right)-\\left(8\\div\\Large\\frac{1}{4}\\normalsize\\right)[\/latex]<\/p>\r\nMultiply [latex] 3\\cdot\\Large\\frac{1}{3}[\/latex] first.\r\n<p style=\"text-align: center;\">[latex]\\left( 3\\cdot\\Large\\frac{1}{3}\\normalsize\\right)-\\left(8\\div\\Large\\frac{1}{4}\\normalsize\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left(1\\right)-\\left(8\\div\\Large\\frac{1}{4}\\normalsize\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Now, divide [latex]8\\div\\Large\\frac{1}{4}[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]8\\div\\Large\\frac{1}{4}\\normalsize =\\Large\\frac{8}{1}\\normalsize\\cdot\\Large\\frac{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\\Large\\frac{1}{3}\\normalsize -8\\div\\Large\\frac{1}{4}\\normalsize =-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<h2>Exponents and Square Roots<\/h2>\r\nIn this section, we expand our skills with applying the order of operation rules to expressions with\u00a0exponents and square roots.\u00a0If 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\r\nRecall 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],\u00a0[latex]7[\/latex]is the base and\u00a0[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\u00a0[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\u00a0[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\r\nWhen 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\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\r\nIn 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]\\Large\\frac{\\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 top and 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, simplify the result and [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=32-11=21[\/latex]<\/p>\r\nNow put the fraction back together to see if any more simplifying needs to be done.\r\n[latex]\\Large\\frac{7}{21}[\/latex], this can be reduced to [latex]\\Large\\frac{1}{3}[\/latex]\r\n<h4>Answer<\/h4>\r\n<p style=\"text-align: center;\">[latex]\\Large\\frac{\\sqrt{7+2}+2^2}{(8)(4)-11}=\\frac{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(\\Large\\frac{1}{2}\\normalsize\\right)}^{2}}+{{\\left(\\Large\\frac{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(\\Large\\frac{1}{2}\\normalsize\\right)^{2}+\\left(\\Large\\frac{1}{4}\\normalsize\\right)^{3}\\cdot32[\/latex]<\/p>\r\nEvaluate: [latex]\\left(\\Large\\frac{1}{2}\\normalsize\\right)^{2}=\\Large\\frac{1}{2}\\normalsize\\cdot\\Large\\frac{1}{2}\\normalsize =\\Large\\frac{1}{4}[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\Large\\frac{1}{4}\\normalsize +\\left(\\Large\\frac{1}{4}\\normalsize\\right)^{3}\\cdot32[\/latex]<\/p>\r\nEvaluate: [latex]\\left(\\Large\\frac{1}{4}\\normalsize\\right)^{3}=\\Large\\frac{1}{4}\\normalsize\\cdot\\Large\\frac{1}{4}\\normalsize\\cdot\\Large\\frac{1}{4}\\normalsize=\\Large\\frac{1}{64}[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\Large\\frac{1}{4}\\normalsize +\\Large\\frac{1}{64}\\normalsize\\cdot32[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex]\\Large\\frac{1}{4}\\normalsize +\\Large\\frac{32}{64}[\/latex]<\/p>\r\nSimplify. [latex]\\Large\\frac{32}{64}\\normalsize =\\Large\\frac{1}{2}[\/latex], so you can add [latex]\\Large\\frac{1}{4}\\normalsize +\\Large\\frac{1}{2}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\Large\\frac{1}{4}\\normalsize +\\Large\\frac{1}{2}\\normalsize =\\Large\\frac{3}{4}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] {{\\left(\\Large\\frac{1}{2}\\normalsize\\right)}^{2}}+{{\\left(\\Large\\frac{1}{4} \\normalsize\\right)}^{3}}\\cdot 32=\\Large\\frac{3}{4}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nSome people use a saying to help them remember the order of operations. This saying is called PEMDAS or <strong>P<\/strong>lease <strong>E<\/strong>xcuse <strong>M<\/strong>y <strong>D<\/strong>ear <strong>A<\/strong>unt <strong>S<\/strong>ally. The first letter of each word begins with the same letter of an arithmetic operation.\r\n\r\n<strong>P<\/strong>lease [latex] \\displaystyle \\Rightarrow [\/latex] <strong>P<\/strong>arentheses (and other grouping symbols)\r\n<strong>E<\/strong>xcuse [latex] \\displaystyle \\Rightarrow [\/latex] <strong>E<\/strong>xponents\r\n<strong>M<\/strong>y <strong>D<\/strong>ear [latex] \\displaystyle \\Rightarrow [\/latex] <strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision (from left to right)\r\n<strong>A<\/strong>unt <strong>S<\/strong>ally [latex] \\displaystyle \\Rightarrow [\/latex] <strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction (from left to right)\r\n\r\n<strong>Note:<\/strong> Even though multiplication comes before division in the saying, division could be performed first. Which is performed first, between multiplication and division, is determined by which comes first when reading from left to right. The same is true of addition and subtraction. Don't let the saying confuse you about this!\r\n\r\nThe order of operations gives us a consistent sequence to use in computation. Without the order of operations, you could come up with different answers to the same computation problem. (Some of the early calculators, and some inexpensive ones, do NOT use the order of operations. In order to use these calculators, the user has to input the numbers in the correct order.)","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Recognize and combine like terms in an expression<\/li>\n<li>Use the order of operations to simplify expressions<\/li>\n<\/ul>\n<\/div>\n<h2><\/h2>\n<p>Some important terminology before we begin:<\/p>\n<ul>\n<li><strong>operations\/operators:<\/strong>\u00a0In mathematics we call things like multiplication, division, addition, and subtraction operations. \u00a0They are the verbs of the math world, doing work on numbers and variables. The symbols used to denote operations are called operators, such as [latex]+{, }-{, }\\times{, }\\div[\/latex]. As you learn more math, you will learn more operators.<\/li>\n<li><strong>term:\u00a0<\/strong>Examples of terms would be [latex]2x[\/latex] and [latex]-\\Large\\frac{3}{2}[\/latex] or [latex]a^3[\/latex]. Even lone integers can be a term, like [latex]0[\/latex].<\/li>\n<li><strong>expression:\u00a0<\/strong>A\u00a0mathematical expression is one that connects terms with mathematical operators.\u00a0For example \u00a0[latex]\\Large\\frac{1}{2}\\normalsize +\\left(2^2\\right)- 9\\div\\Large\\frac{6}{7}[\/latex] is an expression.<\/li>\n<\/ul>\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-1\" 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>Simplified Expression:<\/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-2\" 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>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.<\/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-3\" 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\\Large\\frac{1}{3}\\normalsize -8\\div\\Large\\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]\\left(3\\cdot\\Large\\frac{1}{3}\\normalsize\\right)-\\left(8\\div\\Large\\frac{1}{4}\\normalsize\\right)[\/latex]<\/p>\n<p>Multiply [latex]3\\cdot\\Large\\frac{1}{3}[\/latex] first.<\/p>\n<p style=\"text-align: center;\">[latex]\\left( 3\\cdot\\Large\\frac{1}{3}\\normalsize\\right)-\\left(8\\div\\Large\\frac{1}{4}\\normalsize\\right)[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\left(1\\right)-\\left(8\\div\\Large\\frac{1}{4}\\normalsize\\right)[\/latex]<\/p>\n<p style=\"text-align: left;\">Now, divide [latex]8\\div\\Large\\frac{1}{4}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]8\\div\\Large\\frac{1}{4}\\normalsize =\\Large\\frac{8}{1}\\normalsize\\cdot\\Large\\frac{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\\Large\\frac{1}{3}\\normalsize -8\\div\\Large\\frac{1}{4}\\normalsize =-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-4\" 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<h2>Exponents and Square Roots<\/h2>\n<p>In this section, we expand our skills with applying the order of operation rules to expressions with\u00a0exponents and square roots.\u00a0If 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<p>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],\u00a0[latex]7[\/latex]is the base and\u00a0[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\u00a0[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\u00a0[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-5\" 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<p>When 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<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-6\" 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<p>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]\\Large\\frac{\\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 top and 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, simplify the result and [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=32-11=21[\/latex]<\/p>\n<p>Now put the fraction back together to see if any more simplifying needs to be done.<br \/>\n[latex]\\Large\\frac{7}{21}[\/latex], this can be reduced to [latex]\\Large\\frac{1}{3}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p style=\"text-align: center;\">[latex]\\Large\\frac{\\sqrt{7+2}+2^2}{(8)(4)-11}=\\frac{1}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" 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(\\Large\\frac{1}{2}\\normalsize\\right)}^{2}}+{{\\left(\\Large\\frac{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(\\Large\\frac{1}{2}\\normalsize\\right)^{2}+\\left(\\Large\\frac{1}{4}\\normalsize\\right)^{3}\\cdot32[\/latex]<\/p>\n<p>Evaluate: [latex]\\left(\\Large\\frac{1}{2}\\normalsize\\right)^{2}=\\Large\\frac{1}{2}\\normalsize\\cdot\\Large\\frac{1}{2}\\normalsize =\\Large\\frac{1}{4}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\Large\\frac{1}{4}\\normalsize +\\left(\\Large\\frac{1}{4}\\normalsize\\right)^{3}\\cdot32[\/latex]<\/p>\n<p>Evaluate: [latex]\\left(\\Large\\frac{1}{4}\\normalsize\\right)^{3}=\\Large\\frac{1}{4}\\normalsize\\cdot\\Large\\frac{1}{4}\\normalsize\\cdot\\Large\\frac{1}{4}\\normalsize=\\Large\\frac{1}{64}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\Large\\frac{1}{4}\\normalsize +\\Large\\frac{1}{64}\\normalsize\\cdot32[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\\Large\\frac{1}{4}\\normalsize +\\Large\\frac{32}{64}[\/latex]<\/p>\n<p>Simplify. [latex]\\Large\\frac{32}{64}\\normalsize =\\Large\\frac{1}{2}[\/latex], so you can add [latex]\\Large\\frac{1}{4}\\normalsize +\\Large\\frac{1}{2}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\Large\\frac{1}{4}\\normalsize +\\Large\\frac{1}{2}\\normalsize =\\Large\\frac{3}{4}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]{{\\left(\\Large\\frac{1}{2}\\normalsize\\right)}^{2}}+{{\\left(\\Large\\frac{1}{4} \\normalsize\\right)}^{3}}\\cdot 32=\\Large\\frac{3}{4}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>Some people use a saying to help them remember the order of operations. This saying is called PEMDAS or <strong>P<\/strong>lease <strong>E<\/strong>xcuse <strong>M<\/strong>y <strong>D<\/strong>ear <strong>A<\/strong>unt <strong>S<\/strong>ally. The first letter of each word begins with the same letter of an arithmetic operation.<\/p>\n<p><strong>P<\/strong>lease [latex]\\displaystyle \\Rightarrow[\/latex] <strong>P<\/strong>arentheses (and other grouping symbols)<br \/>\n<strong>E<\/strong>xcuse [latex]\\displaystyle \\Rightarrow[\/latex] <strong>E<\/strong>xponents<br \/>\n<strong>M<\/strong>y <strong>D<\/strong>ear [latex]\\displaystyle \\Rightarrow[\/latex] <strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision (from left to right)<br \/>\n<strong>A<\/strong>unt <strong>S<\/strong>ally [latex]\\displaystyle \\Rightarrow[\/latex] <strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction (from left to right)<\/p>\n<p><strong>Note:<\/strong> Even though multiplication comes before division in the saying, division could be performed first. Which is performed first, between multiplication and division, is determined by which comes first when reading from left to right. The same is true of addition and subtraction. Don&#8217;t let the saying confuse you about this!<\/p>\n<p>The order of operations gives us a consistent sequence to use in computation. Without the order of operations, you could come up with different answers to the same computation problem. (Some of the early calculators, and some inexpensive ones, do NOT use the order of operations. In order to use these calculators, the user has to input the numbers in the correct order.)<\/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-36\">\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>Order of Operations with a Fraction Containing a Square Root. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Provided by<\/strong>: https:\/\/youtu.be\/9suc63qB96o. <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: Combining Like Terms. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <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). <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) . <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^n*b^m. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/JjBBgV7G_Qw\">https:\/\/youtu.be\/JjBBgV7G_Qw<\/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). <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>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\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/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: Combining Like Terms. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/JIleqbO8Tf0\">https:\/\/youtu.be\/JIleqbO8Tf0<\/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>Order of Operations with a Fraction Containing a Square Root. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/9suc63qB96o\">https:\/\/youtu.be\/9suc63qB96o<\/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":395986,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 2: Combining Like Terms\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/b9-7eu29pNM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Simplify an Expression in the Form: 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