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1 Lesson 2.5.4 More on the Order of Operations More on the Order of Operations

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2 Lesson 2.5.4 More on the Order of Operations California Standard: Number Sense 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. What it means for you: You’ll learn how to use the PEMDAS rules with expressions that have decimals, fractions, and exponents. Key words: PEMDAS operations exponent

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3 Lesson 2.5.4 More on the Order of Operations In Chapter One you saw how the order of operations rules help you to figure out which operation you need to do first in a calculation. This Lesson will review what the order is, and give you practice at applying it to expressions with exponents in them.

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4 PEMDAS Tells You What Order to Follow Lesson 2.5.4 More on the Order of Operations When you come across an expression that contains multiple operations, the PEMDAS rule will help you to work out which one to do first. For example: Parentheses4 + 6 (2 + 4) 2 – 10 ÷ 2 = 215 = 4 + 216 – 5 = 4 + 6 36 – 10 ÷ 2 = 4 + 6 (6) 2 – 10 ÷ 2 Addition and Subtraction Multiplication and Division Exponents Addition and subtraction have equal priority too — work them out from left to right. Multiplication and division have equal priority in PEMDAS. You work them out from left to right.

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5 Example 1 Solution follows… Lesson 2.5.4 More on the Order of Operations Evaluate the expression 5 2 – 16 ÷ 2 3 (3 + 2). Solution 5 2 – 16 ÷ 2 3 (3 + 2) Do the addition in the parentheses Then evaluate the two exponents Next it’s multiplication and division — do the division first, as it comes first Then do the multiplication Finally do the subtraction = 5 2 – 16 ÷ 2 3 5 Write out the expression = 25 – 16 ÷ 8 5 = 25 – 2 5 = 25 – 10 = 15

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6 Guided Practice Solution follows… Lesson 2.5.4 More on the Order of Operations Evaluate the expressions in Exercises 1–6. 1. 6 – 10 3 2 2. (5 – 3) 3 + 4 3 ÷ 8 3. 2 4 + (3 2 – 10) 2 4. 5 + 6 4 ÷ (6 – 2) 1 5. (36 ÷ 12 – 2 4 ) 2 6. (10 2 – 5) 2 – (4 ÷ 2) 3 3 6 – 10 9 = 6 – 90 = –84 2 4 + (6 – 10) 2 = 2 4 + (–4) 2 = 16 + 16 = 32 (36 ÷ 12 – 16) 2 = (3 – 16) 2 = (–13) 2 = 169 2 3 + 4 3 ÷ 8 = 8 + 64 ÷ 8 = 8 + 8 = 16 5 + 6 4 ÷ 4 1 = 5 + 1296 ÷ 4 = 5 + 324 = 329 = 15 2 – 2 3 3 = 225 – 8 3 = 225 – 24 = 201

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7 Take Care with Expressions That Have Negative Signs Lesson 2.5.4 More on the Order of Operations When an expression contains a combination of negative numbers and exponents, you need to think carefully about what it means. For example: –(2 2 ) = –(2 2) = –4 (–2) 2 = –2 –2 = 4

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8 Example 2 Solution follows… Lesson 2.5.4 More on the Order of Operations Evaluate the expression (–(3 2 ) 5) + (–3) 2. Solution (–(3 2 ) 5) + (–3) 2 Evaluate the exponent in the inner parentheses Do the multiplication in the parentheses Evaluate the exponent Finally do the addition = (–9 5) + (–3) 2 Write out the expression = –45 + (–3) 2 = –45 + 9 = –36

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9 Guided Practice Solution follows… Lesson 2.5.4 More on the Order of Operations Evaluate the expressions in Exercises 7–12. 7. –(4 2 )8. (–4) 2 9. –(2 2 ) 5 + 110. (–4) 2 ÷ 2 – 4 11. 10 + (2 –(5 2 )) + (–7) 2 12. 12 + (–(2 2 ) + (–2) 2 ) ÷ 2 –(4 4) = –16 –(2 2) 5 + 1 = –4 5 + 1 = –19 10 + (2 –(5 5)) + (–7 –7) = 10 + (2 –25) + 49 = 10 – 50 + 49 = 9 –4 –4 = 16 (–4 –4) ÷ 2 – 4 = (16 ÷ 2) – 4 = 4 12 + (–(2 2) + (–2 –2)) ÷ 2 = 12 + (–4 + 4) ÷ 2 = 12 + 0 ÷ 2 = 12

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10 The Order Applies to Decimals and Fractions Too Lesson 2.5.4 More on the Order of Operations When you’re working out a problem involving decimals or fractions you follow the same order of operations. Parentheses Addition and Subtraction Multiplication and Division Exponents P ASAS MDMD E

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11 Example 3 Solution follows… Lesson 2.5.4 More on the Order of Operations 1 2 1 16 Evaluate the expression + (10 – 7) 2. 4 1 2 1 16 + (10 – 7) 2 4 Do the subtraction in the parentheses Then evaluate the two exponents Solution 1 2 1 16 = + 3 2 4 1 16 = + 9 1 16 Write out the expression Solution continues…

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12 Finally do the addition and simplify Example 3 Lesson 2.5.4 More on the Order of Operations 1 2 1 16 Evaluate the expression + (10 – 7) 2. 4 Solution (continued) Perform the multiplication 1 16 = + 9 1 16 9 = + 1 16 5 8 = 10 16

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13 Example 4 Solution follows… Lesson 2.5.4 More on the Order of Operations Evaluate the expression 0.25 + 7.75 ÷ 3.1 – (0.3) 4. Solution 0.25 + 7.75 ÷ 3.1 – (0.3) 4 Evaluate the exponent Then perform the division Do the addition first, as it comes first Finally do the subtraction = 0.25 + 7.75 ÷ 3.1 – 0.0081 Write out the expression = 0.25 + 2.5 – 0.0081 = 2.75 – 0.0081 = 2.7419

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14 Evaluate the expressions in Exercises 13–16. 13. + – 14. 3 + 4 ÷ – 2 15. 0.1 + (0.25) 2 – 0.2 ÷ 2 16. (0.72 + 0.08) ÷ 16 + (0.4) 2 1 8 1 4 2 1 2 3 3 2 Guided Practice Solution follows… Lesson 2.5.4 More on the Order of Operations 0.1 + (0.25 0.25) – 0.2 ÷ 2 = 0.1 + 0.0625 – 0.1 = 0.0625 (0.72 + 0.08) ÷ 16 + (0.4 0.4) = 0.8 ÷ 16 + 0.16 = 0.05 + 0.16 = 0.21 8 63 2 4 3 – = 8 1 8 64 – 8 1 = 1 2 16 99 16 1 3 + 4 ÷ – 2 = 2 1 16 3 + 8 – 2 = 6 or 16 3

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15 Evaluate the expressions in Exercises 17–20. 17. ÷ + 18. 0.5 (1 + 0.25) 2 + 1.2 19. 2 + (5 ÷ 10) 2 4 20. (5 0.1 + 0.2) 1 2 2 1 5 2 3 4 2 1 2 2 3 3 1 2 Guided Practice Solution follows… Lesson 2.5.4 More on the Order of Operations 0.5 (1.25 1.25) + 1.2 = 0.5 1.5625 + 1.2 = 0.78125 + 1.2 = 1.98125 4 1 2 + (0.5 0.5) 4 = + 0.25 4 = 2 1 2 3 108 247 4 6 2 + = 16 36 += 6 2 3 216 8 250 7 (0.5 + 0.2) = 0.7 5252 1212 25 1 10 7 = 25 1 = = 0.028

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16 Independent Practice Solution follows… Lesson 2.5.4 More on the Order of Operations Evaluate the expressions in Exercises 1–6. 1. 2. (4 2 – 2 3 ) ÷ 2 2 + 8 1 3. (10 + 2 4 3) + (5 2 – 15) 2 4. –3 3 2 2 + 9 5. (–6) 3 3 – 12 2 6. (4 3 – 3 4 ) 2 ÷ (17) 2 12 + 2 3 5 104 –99158 1–792

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17 Independent Practice Solution follows… Lesson 2.5.4 More on the Order of Operations 7. In the expression ( x – y 2 z ) 6, x, y, and z stand for whole numbers. If you evaluate it, will the expression have a positive or a negative value? (The expression is not equal to zero.) Explain your answer. Positive: when you raise any number, positive or negative, to an even power, the result is always positive.

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18 Independent Practice Solution follows… Lesson 2.5.4 More on the Order of Operations Evaluate the expressions in Exercises 8–13. 8. + 2 9. (0.5) 2 + 0.8 ÷ (0.1) 3 10. ÷ + 411. (0.5 + 1.8) 2 1.5 + 0.065 12. 0.5 – 13. (0.2 4 – 0.3) 2 + 2 1 3 2 2 27 2 3 2 4 5 1 6 2 2 6 8 1 4 3 1 2 3 7 800.25 8 1 0.125 or 36 29 8 2 1 0.5 or

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19 Independent Practice Solution follows… Lesson 2.5.4 More on the Order of Operations 14. Lakesha is making bread. She has lb of flour, which she splits into two equal piles. 5 4 lb 16 15 Then she splits each of these in half again. She adds three of the resulting piles to her mixture. How much flour has she added to her mixture? Give your answer as a fraction.

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20 Round Up Lesson 2.5.4 More on the Order of Operations When you have an expression containing exponents, you must follow the order of operations to evaluate it. You use the same order with expressions that contain fractions and decimals too.

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