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Order of Operations
With Exponents |
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Unit 7 |
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Problem:
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Evaluate this arithmetic expression: 18 + 36 ÷ 32
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In the last lesson, we learned how to evaluate an arithmetic expression with more than
one operation according to the following rules:
| Rule 1: |
Simplify all operations inside parentheses. |
| Rule 2: |
Perform all multiplications and divisions, working from left to right. |
| Rule 3: |
Perform all additions and subtractions, working from left to right. |
However, the problem above includes an exponent, so we cannot solve it without
revising our rules.
| Rule 1: |
Simplify all operations inside parentheses. |
| Rule 2: |
Simplify all exponents, working from left to right. |
| Rule 3: |
Perform all multiplications and divisions, working from left to right. |
| Rule 4: |
Perform all additions and subtractions, working from left to right. |
We can solve the problem above using our revised order of operations.
| Problem: |
Evaluate this arithmetic expression 18 + 36 ÷ 32
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| Solution: |
| 18 + 36 ÷ 32 |
= 18 + 36 ÷ 9 |
Simplify all exponents (Rule 2) |
| 18 + 36 ÷ 9 |
= 18 + 4 |
Division (Rule 3) |
| 18 + 4 |
= 22 |
Addition (Rule 4) |
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Let's look at some other examples that involve our new rules for order of operations.
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Example 1:
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Evaluate 52 x 24
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| Solution: |
| 52 x 24 |
= 25 x 24 |
Simplify all exponents, working from left to right (Rule 2) |
| 25 x 24 |
= 25 x 16 |
| 25 x 16 |
= 400 |
Multiplication (Rule 3) |
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Example 2:
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Evaluate 289 - (3 x 5)2
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| Solution: |
| 289 - (3 x 5)2 |
= 289 - 152 |
Simplify all operations inside parentheses (Rule 1) |
| 289 - 152 |
= 289 - 225 |
Simplify all exponents (Rule 2) |
| 289 - 225 |
= 64 |
Subtraction (Rule 4) |
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Example 3:
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Evaluate 8 + (2 x 5) x 34 ÷ 9
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| Solution: |
| 8 + (2 x 5) x 34 ÷ 9 |
= 8 + 10 x 34 ÷ 9 |
Simplify all operations inside parentheses (Rule 1) |
| 8 + 10 x 34 ÷ 9 |
= 8 + 10 x 81 ÷ 9 |
Simplify all exponents (Rule 2) |
| 8 + 10 x 81 ÷ 9 |
= 8 + 810 ÷ 9 |
Perform all multiplications and divisions, working from left to right (Rule 3) |
| 8 + 810 ÷ 9 |
= 8 + 90 |
| 8 + 90 |
= 98 |
Addition (Rule 4) |
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Example 4:
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An interior decorator charges $15 per square foot to lay a carpet, and
an installation fee of $150. If the room is square and each side measures 12 feet,
how much will it cost to carpet it?
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Solution:
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If one side of the square-shaped room is 12 feet, then the area of the room is
(12 feet)2.
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| 15 x 122 + 150 |
= 15 x 144 + 150 |
Simplify all exponents (Rule 2) |
| 15 x 144 + 150 |
= 2,160 + 150 |
Multiplication (Rule 3) |
| 2,160 + 150 |
= 2,310 |
Addition (Rule 4) |
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Answer:
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It will cost $2,310 to carpet this room.
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| Summary: |
To help us remember the order of operations, we can use the mnemonic PEMDAS,
which stands for:
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| Please |
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Excuse |
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My |
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Dear |
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Aunt |
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Sally |
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| Parentheses, |
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Exponents, |
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Multiplication & Division, |
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Addition & Subtraction |
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Note that although there are six words, they correspond to four rules.
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Exercises
Directions: Complete each exercise by applying the rules for order of operations. Click once in an
ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to
indicate whether your answer is correct or incorrect. To start over, click CLEAR.
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1.
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32 x 43
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2.
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27 - 256 ÷ 43
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3.
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9 x (5 + 3)2 - 144
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4.
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7 + 3 x 24 ÷ 6
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5.
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A carpenter charges $10 per square foot to lay a floor. If a square-shaped
hallway is 6 feet along one side, and the customer has a coupon for $25 off the total,
then how much will the floor cost?
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