#set document(title: "3.3 Order of Operations", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 3.3#h(0.6em)Order of Operations #figure(figph[A close-up view of a notebook show a mathematical calculation. A calculator and a pen are placed by the notebook.], alt: "A close-up view of a notebook show a mathematical calculation. A calculator and a pen are placed by the notebook.", caption: [Calculators may automatically apply order of operations to calculations.]) After completing this module, you should be able to: + Simplify expressions using order of operations. + Simplify expressions using order of operations involving grouping symbols. Calculates else sure someone be rules expect explicit we what that needs to need make, we to that them what be calculate to calculated first. You probably read that sentence and couldn't make heads or tails of it. Seems like it might concern calculations, but maybe it concerns needs? You may even be attempting to unscramble the sentence as you read it, placing words in the order you might expect them to appear in. The reason that the sentence makes no sense is that the words don't follow the order you expect them to follow. Unscrambled, the sentence was intended to be “To be sure that someone else calculates when we expect them to calculate, we need rules that make explicit what needs to be calculated first.” Similarly, when working with math expressions and equations, if we don't follow the rules for #strong[order of operations], arithmetic expressions make no sense. Just a simple expression would be problematic if we didn't have some rules to tell us what to calculate first. For instance, #math.equation(block: false, alt: "4 times 2 squared plus 3 plus 5 squared")[$4 × 2^(2) + 3 + 5^(2)$] can be calculated in many ways. You could get 5,184. Or, you could get 80. Or, 96. The issue is that without following a set of rules for calculation, the same expression will give various results. In case you are curious, using the appropriate order of operations, we find #math.equation(block: false, alt: "4 times 2 squared plus 3 plus 5 squared equals 44")[$4 × 2^(2) + 3 + 5^(2) = 44$]. === Simplify Expressions Using Order of Operations The order in which mathematical operations is performed is a convention that makes it easier for anyone to correctly calculate. They follow the acronym EMDAS: #figure(table( columns: 2, align: left, inset: 6pt, table.header([E], [Exponents]), [M/D], [Multiplication and division], [A/S], [Addition and Subtraction], )) So, what does EMDAS tell us to do? In an equation, moving left to right, we begin by calculating all the exponents first. Once the exponents have been calculated, we again move left to right, calculating the multiplications and divisions, one at a time. Multiplication and division hold the same position in the ordering, so when you encounter one or the other at this step, do it. Once the multiplications and divisions have been calculated, we again move left to right, calculating the additions and subtractions, one at a time. Additions and subtractions hold the same position in the ordering, so when you encounter one or the other at this step, do it. (You may have previously learned the order of operations as PEMDAS, with parentheses first; we will add that aspect later on.) We’ll explore this as we work an example. #examplebox("Example 1")[Using Two Order of Operations][ Calculate #math.equation(block: false, alt: "21 minus 4 times 13")[$21 − 4 × 13$]. #solutionbox[ There are no exponents in this expression, so the next operations to check are multiplication and division. #strong[Step 1:] Moving left to right, the first multiplication encountered is 4 multiplied by 13. We perform that operation first. #math.equation(block: true, alt: "21 minus 4 times 13; equals 21 minus 52")[$21 − 4 × 13 \ = 21 − 52$] #strong[Step 2:] The only operation remaining is the subtraction. #math.equation(block: true, alt: "21 minus 52 equals minus 31")[$21 − 52 = − 31$]. So, #math.equation(block: false, alt: "21 minus 4 times 13 equals minus 31")[$21 − 4 × 13 = − 31$]. ] ] #examplebox("Example 2")[Using Two Order of Operations][ Calculate #math.equation(block: false, alt: "4 times 8 cubed")[$4 × 8^(3)$]. #solutionbox[ #strong[Step 1:] Moving left to right, we see there is an exponent. We calculate the exponent first. #math.equation(block: true, alt: "4 times 8 cubed equals 4 times open parenthesis 8 times 8 times 8 close parenthesis equals 4 times 512")[$4 × 8^(3) = 4 × ( 8 × 8 × 8 ) = 4 × 512$] #strong[Step 2:] The only operation remaining is the multiplication. #math.equation(block: true, alt: "4 times 512 equals 2,048")[$4 × 512 = 2,048$] So, #math.equation(block: false, alt: "4 times 8 cubed equals 2,048")[$4 × 8^(3) = 2,048$]. ] ] #examplebox("Example 3")[Using Three Order of Operations][ Calculate #math.equation(block: false, alt: "2 plus 3 squared times 4")[$2 + 3^(2) × 4$]. #solutionbox[ #strong[Step 1:] To calculate this, move left to right, and compute all the exponents first. The only exponent we see is the squaring of the 3, so that is calculated first. #math.equation(block: true, alt: "2 plus 3 squared times 4 equals 2 plus open parenthesis 3 times 3 close parenthesis times 4 equals 2 plus 9 times 4")[$2 + 3^(2) × 4 = 2 + ( 3 × 3 ) × 4 = 2 + 9 × 4$] #strong[Step 2:] Since the exponents are all calculated, now calculate all the multiplications and divisions moving left to right. The only multiplication or division present is 9 times 4. #math.equation(block: true, alt: "2 plus 9 times 4 equals 2 plus 36")[$2 + 9 × 4 = 2 + 36$] #strong[Step 3:] Moving left to right, perform the additions and subtractions. There is only one such operation, 2 plus 36. #math.equation(block: true, alt: "2 plus 36 equals 38")[$2 + 36 = 38$] So, #math.equation(block: false, alt: "2 plus 3 squared times 4 equals 38")[$2 + 3^(2) × 4 = 38$]. ] ] Even if the expression being calculated gets more complicated, we perform the operations in the order: EMDAS. #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Order_of_Operations_1")[Order of Operations 1] ] #examplebox("Example 4")[Using Eight Order of Operations][ Correctly apply the order of operations to compute the following: #math.equation(block: true, alt: "4 minus 25 times 6 / 10 times 3 squared plus 7 times 2 cubed")[$4 − 25 × 6 / 10 × 3^(2) + 7 × 2^(3)$]. #solutionbox[ #strong[Step 1:] To do so, calculate the exponents first, moving left to right. There are two occurrences of exponents in the expression, 3 squared and 2 cubed. #math.equation(block: true, alt: "4 minus 25 times 6 / 10 times 3 squared plus 7 times 2 cubed equals 4 minus 25 times 6 / 10 times 9 plus 7 times 8")[$4 − 25 × 6 / 10 × 3^(2) + 7 × 2^(3) = 4 − 25 × 6 / 10 × 9 + 7 × 8$] #strong[Step 2:] Now that the exponents are calculated, perform the multiplication and division, moving left to right. The first is the product of 25 and 6. #math.equation(block: true, alt: "4 minus 25 times 6 / 10 times 9 plus 7 times 8; equals 4 minus 150 / 10 times 9 plus 7 times 8")[$4 − 25 × 6 / 10 × 9 + 7 × 8 \ = 4 − 150 / 10 × 9 + 7 × 8$] #strong[Step 3:] Next is the 150 divided by 10. #math.equation(block: true, alt: "4 minus 150 / 10 times 9 plus 7 times 8; equals 4 minus 15 times 9 plus 7 times 8")[$4 − 150 / 10 × 9 + 7 × 8 \ = 4 − 15 × 9 + 7 × 8$] #strong[Step 4:] Next is 15 multiplied by 9. #math.equation(block: true, alt: "4 minus 15 times 9 plus 7 times 8; equals 4 minus 135 plus 7 times 8")[$4 − 15 × 9 + 7 × 8 \ = 4 − 135 + 7 × 8$] #strong[Step 5:] Finally, multiply the 7 and 8. #math.equation(block: true, alt: "4 minus 135 plus 7 times 8; equals 4 minus 135 plus 56")[$4 − 135 + 7 × 8 \ = 4 − 135 + 56$] As all the multiplications and divisions have been calculated, the additions and subtractions are performed, moving left to right. #math.equation(block: true, alt: "4 minus 135 plus 56; equals minus 131 plus 56; equals minus 131 plus 56; equals minus 75")[$4 − 135 + 56 \ = − 131 + 56 \ = − 131 + 56 \ = − 75$] The computed value is −75. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Order_of_Operations_2")[Order of Operations 2] ] #examplebox("Example 5")[Using Six Order of Operations][ Correctly apply the rules for the order of operations to accurately compute the following: #math.equation(block: true, alt: "10 minus 3 times 5 cubed / 15 plus 56 / 4")[$10 − 3 × 5^(3) / 15 + 56 / 4$]. #solutionbox[ #strong[Step 1:] Calculate exponents first, moving left to right: #math.equation(block: true, alt: "10 minus 3 times 5 cubed / 15 plus 56 / 4; equals 10 minus 3 times 125 / 15 plus 56 / 4")[$10 − 3 × 5^(3) / 15 + 56 / 4 \ = 10 − 3 × 125 / 15 + 56 / 4$] #strong[Step 2:] Multiply and divide, moving left to right: #math.equation(block: true, alt: "10 minus 3 times 125 / 15 plus 56 / 4; equals 10 minus 375 / 15 plus 56 / 4; equals 10 minus 25 plus 56 / 4; equals 10 minus 25 plus 14")[$10 − 3 × 125 / 15 + 56 / 4 \ = 10 − 375 / 15 + 56 / 4 \ = 10 − 25 + 56 / 4 \ = 10 − 25 + 14$] #strong[Step 3:] Add and subtract, moving left to right: #math.equation(block: true, alt: "10 minus 25 plus 14; equals minus 15 plus 14; equals minus 1")[$10 − 25 + 14 \ = − 15 + 14 \ = − 1$] ] ] #examplebox("Example 6")[Using Order of Operations][ Correctly apply the rules for the order of operations to accurately compute the following: #math.equation(block: false, alt: "open parenthesis minus 8 close parenthesis / 2 times 3 minus 9 times 2 to the power 4 / 12 plus 9 times open parenthesis minus 4 close parenthesis squared / 2 cubed")[$( − 8 ) / 2 × 3 − 9 × 2^(4) / 12 + 9 × attach(( − 4 ), t: 2) / 2^(3)$]. #solutionbox[ #strong[Step 1:] Calculate the exponents first, moving left to right: #math.equation(block: true, alt: "open parenthesis minus 8 close parenthesis / 2 times 3 minus 9 times 2 to the power 4 / 12 plus 9 times open parenthesis minus 4 close parenthesis squared / 2 cubed; equals open parenthesis minus 8 close parenthesis / 2 times 3 minus 9 times 16 / 12 plus 9 times 12 squared / 2 cubed; equals open parenthesis minus 8 close parenthesis / 2 times 3 minus 9 times 16 / 12 plus 9 times 144 / 2 cubed; equals open parenthesis minus 8 close parenthesis / 2 times 3 minus 9 times 16 / 12 plus 9 times 144 / 8")[$( − 8 ) / 2 × 3 − 9 × 2^(4) / 12 + 9 × attach(( − 4 ), t: 2) / 2^(3) \ = ( − 8 ) / 2 × 3 − 9 × 16 / 12 + 9 × 12^(2) / 2^(3) \ = ( − 8 ) / 2 × 3 − 9 × 16 / 12 + 9 × 144 / 2^(3) \ = ( − 8 ) / 2 × 3 − 9 × 16 / 12 + 9 × 144 / 8$] #strong[Step 2:] Multiply and divide, moving left to right: #math.equation(block: true, alt: "equals open parenthesis minus 8 close parenthesis / 2 times 3 minus 9 times 16 / 12 plus 9 times 144 / 8; equals open parenthesis minus 4 close parenthesis times 3 minus 9 times 16 / 12 plus 9 times 144 / 8; equals open parenthesis minus 12 close parenthesis minus 9 times 16 / 12 plus 9 times 144 / 8; equals open parenthesis minus 12 close parenthesis minus 144 / 12 plus 9 times 144 / 8; equals open parenthesis minus 12 close parenthesis minus 12 plus 9 times 144 / 8; equals open parenthesis minus 12 close parenthesis minus 12 plus 1,296 / 8; equals open parenthesis minus 12 close parenthesis minus 12 plus 162")[$= ( − 8 ) / 2 × 3 − 9 × 16 / 12 + 9 × 144 / 8 \ = ( − 4 ) × 3 − 9 × 16 / 12 + 9 × 144 / 8 \ = ( − 12 ) − 9 × 16 / 12 + 9 × 144 / 8 \ = ( − 12 ) − 144 / 12 + 9 × 144 / 8 \ = ( − 12 ) − 12 + 9 × 144 / 8 \ = ( − 12 ) − 12 + 1,296 / 8 \ = ( − 12 ) − 12 + 162$] #strong[Step 3:] Add and subtract, moving left to right: #math.equation(block: true, alt: "equals open parenthesis minus 12 close parenthesis minus 12 plus 162 equals open parenthesis minus 24 close parenthesis plus 162 equals 138")[$= ( − 12 ) − 12 + 162 = ( − 24 ) + 162 = 138$] ] ] === Using the Order of Operations Involving Grouping Symbols We have examined how to use the order of operations, denoted by EMDAS, to correctly calculate expressions. However, there may be expressions where a multiplication should happen before an exponent, or a subtraction before a division. To indicate an operation should be performed out of order, the operation is placed inside parentheses. When parentheses are present, the operations inside the parentheses are performed first. Adding the parentheses to our list, we now have PEMDAS, as shown below. #figure(table( columns: 3, align: left, inset: 6pt, table.header([P], [Parentheses], []), [E], [Exponents], [], [M/D], [Multiplication and division], [(division is just the multiplication by the reciprocal)], [A/S], [Addition and subtraction], [(subtraction is just the addition of the negative)], )) As said previously, parentheses indicate that some operation or operations will be performed outside the standard order of operation rules. For instance, perhaps you want to multiply 4 and 7 before squaring. To indicate that the multiplication happens before the exponent, the multiplication is placed inside parentheses: #math.equation(block: false, alt: "open parenthesis 4 times 7 close parenthesis squared")[$attach(( 4 × 7 ), t: 2)$]. This means operations inside the parentheses take precedence, or happen before other operations. Now, the first step in calculating arithmetic expressions using the order of operations is to perform operations inside parentheses first. Inside the parentheses, you follow the order of operation rules EMDAS. #examplebox("Example 7")[Prioritizing Parentheses in the Order of Operations][ Correctly apply the rules for the order of operations to accurately compute the following: #math.equation(block: true, alt: "open parenthesis 10 minus 3 close parenthesis times 5 cubed")[$( 10 − 3 ) × 5^(3)$]. #solutionbox[ #strong[Step 1:] Perform all calculations within the parentheses before all other operations. #math.equation(block: true, alt: "open parenthesis 10 minus 3 close parenthesis times 5 cubed equals 7 times 5 cubed")[$( 10 − 3 ) × 5^(3) = 7 × 5^(3)$] #strong[Step 2:] Since all parentheses have been cleared, move left to right, and compute all the exponents next. #math.equation(block: true, alt: "7 times 5 cubed equals 7 times 125")[$7 × 5^(3) = 7 × 125$] #strong[Step 3:] Perform all multiplications and divisions moving left to right. #math.equation(block: true, alt: "7 times 125 equals 875")[$7 × 125 = 875$] ] ] Be aware that there can be more than one set of parentheses, and parentheses within parentheses. When one set of parentheses is inside another set, do the innermost set first, and then work outward. #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Order_of_Operations_3")[Order of Operations 3] ] #examplebox("Example 8")[Working Innermost Parentheses in the Order of Operations][ Correctly apply the rules for order of operations to accurately compute the following: #math.equation(block: true, alt: "4 plus 2 times open parenthesis 3 squared minus open parenthesis 2 plus 5 close parenthesis squared times 4 close parenthesis / open parenthesis 3 plus 8 close parenthesis")[$4 + 2 × ( 3^(2) − attach(( 2 + 5 ), t: 2) × 4 ) / ( 3 + 8 )$]. #solutionbox[ #strong[Step 1:] Perform all calculations within the parentheses before other operations. Evaluate the innermost parentheses first. We can work separate parentheses expressions at the same time. The innermost set of parentheses has the 2 + 5 inside. The 3 + 8 is in a separate set of parentheses, so that addition can occur at the same time as the 2 + 5. #math.equation(block: true, alt: "4 plus 2 times open parenthesis 3 squared minus open parenthesis 2 plus 5 close parenthesis squared times 4 close parenthesis / open parenthesis 3 plus 8 close parenthesis; equals 4 plus 2 times open parenthesis 3 squared minus open parenthesis 7 close parenthesis squared times 4 close parenthesis / open parenthesis 11 close parenthesis")[$4 + 2 × ( 3^(2) − attach(( 2 + 5 ), t: 2) × 4 ) / ( 3 + 8 ) \ = 4 + 2 × ( 3^(2) − attach(( 7 ), t: 2) × 4 ) / ( 11 )$] #strong[Step 2:] Now that those parentheses have been handled, move on to the next set of parentheses. Applying the order of operation rules inside that set of parentheses, the exponent is evaluated first, then the multiplication, and then the addition. #math.equation(block: true, alt: "4 plus 2 times open parenthesis 3 squared minus 7 squared times 4 close parenthesis / 11; equals 4 plus 2 times open parenthesis 3 squared minus 49 times 4 close parenthesis / 11; equals 4 plus 2 times open parenthesis minus 187 close parenthesis / 11")[$4 + 2 × ( 3^(2) − 7^(2) × 4 ) / 11 \ = 4 + 2 × ( 3^(2) − 49 × 4 ) / 11 \ = 4 + 2 × ( − 187 ) / 11$] #strong[Step 3:] Since all parentheses have been cleared, apply the EMDAS rules to finish the calculation. #math.equation(block: true, alt: "4 plus 2 times open parenthesis minus 187 close parenthesis / 11 equals 4 minus 374 / 11 equals 4 minus 34 equals minus 30")[$4 + 2 × ( − 187 ) / 11 = 4 − 374 / 11 = 4 − 34 = − 30$] ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Order_of_Operations_4")[Order of Operations 4] ] === Key Terms - order of operations - PEMDAS === Key Concepts - Establishing shared rules on which arithmetic operations are calculated first is necessary. Without them, different people may find different values for the same expression. - The highest precedence is with expressions in parentheses. This allows parts of an expression to be calculated in an order different than the basic order of operations. - The lowest precedence is addition and subtraction, as they are the basis for all other calculations. - Multiplication and division have precedence over addition and subtraction, as they are representations of repeated addition or subtraction. - Exponents have precedence over multiplication and division, as they represent repeated multiplication and division. === Videos - #link("https://openstax.org/r/Order_of_Operations_1")[Order of Operations 1] - #link("https://openstax.org/r/Order_of_Operations_2")[Order of Operations 2] - #link("https://openstax.org/r/Order_of_Operations_3")[Order of Operations 3] - #link("https://openstax.org/r/Order_of_Operations_4")[Order of Operations 4]