1. Objective
Use the order of operations to simplify expressions.

2. Vocabulary
order of operations

3. When a numerical or algebraic expression contains
more than one operation symbol, the order of
operations tells which operation to perform first.
Order of Operations
Perform operations inside grouping
symbols.
First:
Second:
Evaluate powers.
Third:
Perform multiplication and division
from left to right.
Perform addition and subtraction from
left to right.
Fourth:

4. Grouping symbols include parentheses ( ), brackets [ ], and braces { }
Grouping symbols include parentheses ( ), brackets [ ], and braces { }. If an expression contains more than one set of grouping symbols, evaluate the expression from the innermost set first.

5. Helpful Hint The first letter of these words can help you
remember the order of operations.
Please
Excuse My
Dear
Aunt
Sally
Parentheses
Exponents
Multiply
Divide
Add
Subtract

6. Example 1: Translating from Algebra to Words
Simplify each expression.
A. 15 – 2 · 3 + 1
15 – 2 · 3 + 1
There are no grouping symbols.
15 – 6 + 1
Multiply. 10
Subtract and add from left to right.
B. 12 – ÷ 2
12 – ÷ 2
There are no grouping symbols.
12 – ÷ 2
Evaluate powers. The exponent applies only to the 3.
12 – 9 + 5
Divide.
Subtract and add from left to
right. 8

7. Simplify the expression.
Check It Out! Example 1a
Simplify the expression.
1 2
8 ÷ · 3

8. Check It Out! Example 1b
Simplify the expression.
5.4 –

9. Check It Out! Example 1c
Simplify the expression.
–20 ÷ [–2(4 + 1)]

10. Example 2A: Evaluating Algebraic Expressions
Evaluate the expression for the given value
of x.
10 – x · 6 for x = 3
10 – x · 6
First substitute 3 for x.
10 – 3 · 6
Multiply.
10 – 18
Subtract. –8

11. Example 2B: Evaluating Algebraic Expressions
Evaluate the expression for the given value of x.
42(x + 3) for x = –2
42(x + 3)
First substitute –2 for x.
42(–2 + 3)
Perform the operation inside the parentheses.
42(1)
16(1)
Evaluate powers. 16
Multiply.

12. Check It Out! Example 2a
Evaluate the expression for the given value of x.
14 + x2 ÷ 4 for x = 2

13. Check It Out! Example 2b
Evaluate the expression for the given value of x.
(x · 22) ÷ (2 + 6) for x = 6

14. Fraction bars, radical symbols, and absolute-value symbols can also be used as grouping symbols. Remember that a fraction bar indicates division.

15. Example 3A: Simplifying Expressions with Other Grouping Symbols
2(–4) + 22
42 – 9
The fraction bar acts as a grouping symbol. Simplify the numerator and the denominator before dividing.
2(–4) + 22
42 – 9
–8 + 22
42 – 9
Multiply to simplify the numerator.
–8 + 22
16 – 9
Evaluate the power in the denominator.
Add to simplify the numerator. Subtract to simplify the denominator. 14 7 2
Divide.

16. Example 3B: Simplifying Expressions with Other Grouping Symbols
3| ÷ 2|
The absolute-value symbols act as grouping symbols.
3| ÷ 2|
Evaluate the power.
3| ÷ 2|
Divide within the absolute-value symbols.
3|16 + 4|
3|20|
Add within the absolute-symbols.
3 · 20
Write the absolute value of 20. 60
Multiply.

17. Check It Out! Example 3a
Simplify.
5 + 2(–8)
(–2) – 3 3

18. Check It Out! Example 3b
Simplify.
|4 – 7|2 ÷ –3

19. Check It Out! Example 3c
Simplify.

20. You may need grouping symbols when translating from words to numerical expressions.
Remember!
Look for words that imply mathematical operations.
difference subtract
sum add
product multiply
quotient divide

21. Example 4: Translating from Words to Math
Translate each word phrase into a numerical or algebraic expression.
A. the sum of the quotient of 12 and –3 and the square root of 25
Show the quotient being added to
B. the difference of y and the product of 4
and
Use parentheses so that the product is evaluated first.

22. Check It Out! Example 4
Translate the word phrase into a numerical or algebraic expression: the product of 6.2 and the sum of 9.4 and 8.

23. Example 5: Retail Application
A shop offers gift-wrapping services at three price levels. The amount of money collected for wrapping gifts on a given day can be found by using the expression 2B + 4S + 7D. On Friday the shop wrapped 10 Basic packages B, 6 Super packages S, and 5 Deluxe packages D. Use the expression to find the amount of money collected for gift wrapping on Friday.

24. Example 5 Continued
2B + 4S + 7D
First substitute the value for each variable.
2(10) + 4(6) + 7(5)
Multiply.
Add from left to right. 79
Add.
The shop collected $79 for gift wrapping on Friday.

25. Check It Out! Example 5
Another formula for a player's total number of bases is Hits + D + 2T + 3H. Use this expression to find Hank Aaron's total bases for 1959, when he had 223 hits, 46 doubles, 7 triples, and 39 home runs.

26. Lesson Quiz
Simply each expression. 2.
52 – (5 + 4)
|4 – 8|
1. 2[5 ÷ (–6 – 4)]
3. 5  8 – ÷ 22
Translate each word phrase into a numerical or algebraic expression.
4. 3 three times the sum of –5 and n
5. the quotient of the difference of 34 and 9 and the square root of 25
6. the volume of a storage box can be found using the expression l · w(w + 2). Find the volume of the box if l = 3 feet and w = 2 feet.