1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 1.6 Exponents and Order of Operations

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Understand the terms base and exponent. o Know how to evaluate expressions containing exponents. o Understand how to evaluate expressions with 1 and 0 as exponents. o Know the rules for order of operations.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Understanding the Terms Base and Exponent In each exponential expression, identify the base and the exponent. a. 6 2 b. 10 4 6 is the base, and 2 is the exponent. 10 is the base, and 4 is the exponent.

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Reading Equations Containing Exponential Expressions is read “eight squared is equal to sixty-four” is read “six cubed is equal to two hundred sixteen” is read “five to the fourth power is equal to six hundred twenty-five” a.8 2 = 64 b.6 3 = 216 c.5 4 = 625

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Rewrite each expression in exponential form and then evaluate the expression. a.7 ⋅ 7 Solution Example 3: Writing and Evaluating Exponential Expressions

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. b. 2 ⋅ 2 ⋅ 2 Solution Example 3: Writing and Evaluating Exponential Expressions (cont.)

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Rewrite each expression in exponential form and then evaluate the expression. c. 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 Solution Example 3: Writing and Evaluating Exponential Expressions (cont.)

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Understand the Terms Base and Exponent Common Error Wrong Solution DO NOT multiply the base and the exponent. Correct Solution DO multiply the base by itself.

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Exponents 1 and 0 The Exponent 1 For any number a, a 1 = a. For example,

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Exponents 1 and 0 The Exponent 0 For any nonzero number a, a 0 = 1. For example, Note: The expression 0 0 is undefined.

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Exponents 1 and 0 Notes To help in understanding the exponent 0, note the following pattern of powers of the base 3: Each value is the previous value divided by 3. In this way defining 3 0 = 1 makes sense. (Try this idea with bases other than 3.)

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Evaluating Exponential Expressions Evaluate the following exponential expressions. a.9 1 b.8 0 c.15 0 d.10 1 Solution a.9 1 b.8 0 c.15 0 d.10 1 = 9 = 1 = 10

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Rules for Order of Operations 1.First, simplify within grouping symbols, such as parentheses ( ), brackets [ ], or braces { }. Start with the innermost group. 2.Second, evaluate any numbers or expressions indicated by exponents. 3.Third, moving from left to right, perform any multiplication or division in the order in which it appears. 4.Fourth, moving from left to right, perform any addition or subtraction in the order in which it appears.

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Know the Rules for Order of Operations Notes Note that in Rule 3, neither multiplication nor division has priority over the other. Whichever of these operations occurs first, moving left to right, is done first. In Rule 4, addition and subtraction are handled in the same way. Unless they occur within grouping symbols, addition and subtraction are the last operations to be performed.

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. = 8 − 5 Example 5: Order of Operations Evaluate 14 ÷ 7 + 3 ⋅ 2 − 5. Solution In this example there are no grouping symbols or exponents, so we begin with multiplication and division (left to right). Divide before multiplying in this case. Multiply before adding or subtracting. Add before subtracting in this case. Subtract. 14 ÷ 7 + 3 ⋅ 2 − 5 = 2 + 3 ⋅ 2 − 5 = 2 + 6 − 5 = 3

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Order of Operations Evaluate (6 + 2) + (8 + 1) ÷ 9. Solution (6 + 2) + (8 + 1) ÷ 9 = 8 + 9  9 = 8 + 1 = 9 Operate within parentheses. Divide. Add.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7: Order of Operations Evaluate 2 ⋅ 3 2 + 18 ÷ 3 2. Solution 2 ⋅ 3 2 + 18 ÷ 3 2 = 2 ⋅ 9 + 18  9 = 18 + 2 = 20 Evaluate the exponents. Multiply and divide. Add.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Evaluate 30 ÷ 10 ⋅ 2 3 + 3(6 − 2). Solution = 24 + 12 Example 8: Order of Operations 30 ÷ 10 ⋅ 2 3 + 3(6 − 2) = 30 ÷ 10 ⋅ 2 3 + 3 (4) = 30 ÷ 10 ⋅ 8 + 3 (4) = 36 = 3 ⋅ 8 + 3 (4) Operate within parentheses. Evaluate the exponent. Divide. Multiply in each part. Add.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 9: Order of Operations Evaluate 2[5 2 + (2 ⋅ 3 2 − 10)]. Solution 2[5 2 + (2 ⋅ 3 2 − 10)] = 2[25 + (2 ⋅ 9 − 10)] = 2[25 + (18 − 10)] = 2[25 + 8] = 2 [33] = 66 Evaluate the exponents. Multiply inside the parentheses. Subtract inside the parentheses. Add inside the brackets. Multiply.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 10: Order of Operations Evaluate 3(2 + 2 2 )− 6 − 3 ⋅ 2 2. Solution 44 612 18 12 0

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Find the value of each expression. 1.5 2 2. 3 4 3. 4 0 4. 1 5 Use the rules for order of operations to evaluate each expression. 5.14 ÷ 7 + 3 ⋅ 2 3 6. 9 ÷ 3 ⋅ 2 + 3(6 − 2 2 ) 7.6[(6 −1) 2 − (4 2 −9)]

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1.25 2.81 3.1 4.1 5.26 6.12 7.108