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Contents of parentheses
An expression in an exponent (a small, raised number indicating a power) groups that expression like parentheses do. Evaluate any superscripted expression down to a single number before evaluating the power. In other words, to find 53–1, you can pretend 3 – 1 is in parentheses, making the problem 5(3–1) = 52 = 25.
A few other symbols that you may be familiar with also group expressions together just like parentheses. These include the square root symbol and absolute value bars.
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Powers from left to right
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Multiplication and division from left to right
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Addition and subtraction from left to right
Sample questions
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Evaluate (8 + 62) / (23 – 4).
11. Begin by evaluating the contents of the first set of parentheses. Inside this set, evaluate the power first and do the addition next:
(8 + 62) / (23 – 4)
= (8 + 36) / (23 – 4)
= 44 / (23 – 4)
Move to the next set of parentheses, evaluating the power first and then the subtraction:
= 44 / (8 – 4) = 44 / 4
Finish up by evaluating the division: 44 / 4 = 11.
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Find the value of –1 + (–20 + 33)2.
48. When the entire contents of a set of parentheses is raised to a power, evaluate what’s inside the parentheses before evaluating the power. Inside this set, evaluate the power first and the addition next:
–1 + (–20 + 33)2 = –1 + (–20 + 27)2 = –1 + 72
Next, evaluate the power 72 = 7 x 7 = 49:
= –1 + 49
Finish up by evaluating the addition: –1 + 49 = 48
Practice questions
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Find (62 – 12) / (16 / 23).
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Evaluate –10 – (2 + 32 x –4).
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72 – (3 + 32 / –9)5 = ?
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What is (10 – 114 x 8)4/4+5
Following are the answers to the practice questions:
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(62 – 12) / (16 / 23) = 12.
Focusing on the contents of the first set of parentheses, evaluate the power and then the subtraction:
(62 – 12) / (16 / 23)
= (36 – 12) / (16 / 23)
= 24 / (16 / 23)
Next, work inside the second set of parentheses, evaluating the power first and then the division:
= 24 / (16 / 8) = 24 / 2
Finish by evaluating the division:
= 24 / 2 = 12
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–10 – (2 + 32 x –4) = 24.
Focusing on the contents of the parentheses, evaluate the power first, then the multiplication, and then the addition:
–10 – (2 + 32 x –4) = –10 – (2 + 9 x – 4) = –10 – (2 + –36) = –10 – (–34)
Finish by evaluating the subtraction:
–10 – (–34) = 24
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72 – (3 + 32 / –9)5 = 17.
Focusing inside the parentheses, evaluate the power first, then the division, and then the addition:
72 – (3 + 32 / –9)5
=72 – (3 + 9 / –9)5
=72 – (3 + –1)5
=72 – 25
Next, evaluate both powers in order:
= 49 – 25 = 49 – 32
To finish, evaluate the subtraction:
49 – 32 = 17
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(10 – 114 x 8)4/4+5 = 64.
Focusing inside the first set of parentheses, evaluate the power first, then the multiplication, and then the subtraction:
(10 – 114 x 8)4/4+5 = (10 – 1 x 8)4/4+5 = (10 – 8)4/4+5 = 24/4+5
Next, handle the expression in the exponent, evaluating the division first and then the addition:
21+5 = 26
To finish, evaluate the power:
26 = 64