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Section 1 Part 1 Chapter 5

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1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives 2 6 5 3 4 Integer Exponents – Part 1 Use the product rule for exponents. Define 0 and negative exponents. Use the quotient rule for exponents. 5.1

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. We use exponents to write products of repeated factors. For example, 2 5 is defined as 2 2 2 2 2 = 32. The number 5, the exponent, shows that the base 2 appears as a factor five times. The quantity 2 5 is called an exponential or a power. We read 2 5 as “2 to the fifth power” or “2 to the fifth.” Integer Exponents Slide 5.1- 3

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the product rule for exponents. Objective 1 Slide 5.1- 4

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Product Rule for Exponents If m and n are natural numbers and a is any real number, then a m a n = a m + n. That is, when multiplying powers of like bases, keep the same base and add the exponents. Slide 5.1- 5 Use the product rule for exponents. Be careful not to multiply the bases. Keep the same base and add the exponents.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Apply the product rule, if possible, in each case. a) m 8 m 6 b) m 5 p 4 c) (–5p 4 ) (–9p 5 ) d) (–3x 2 y 3 ) (7xy 4 ) = m 8+6 = m 14 Cannot be simplified further because the bases m and p are not the same. The product rule does not apply. = 45p 9 = (–5)(–9)(p 4 p 5 )= 45p 4+5 = –21x 3 y 7 = (–3)(7) x 2 xy 3 y 4 = –21x 2+1 y 3+4 Slide 5.1- 6 CLASSROOM EXAMPLE 1 Using the Product Rule for Exponents Solution:

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Define 0 and negative exponents. Objective 2 Slide 5.1- 7

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Zero Exponent If a is any nonzero real number, then a 0 = 1. Slide 5.1- 8 Define 0 and negative exponents. The expression 0 0 is undefined.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Evaluate. 29 0 (–29) 0 –29 0 8 0 – 15 0 = 1 = – (29 0 ) = –1 = 1 – 1 = 0 Slide 5.1- 9 CLASSROOM EXAMPLE 2 Using 0 as an Exponent Solution:

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Negative Exponent For any natural number n and any nonzero real number a, A negative exponent does not indicate a negative number; negative exponents lead to reciprocals. Slide 5.1- 10 Define 0 and negative exponents.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write with only positive exponents. 6 -5 (2x) -4, x ≠ 0 –7p -4, p ≠ 0 Evaluate 4 -1 – 2 -1. Slide 5.1- 11 CLASSROOM EXAMPLE 3 Using Negative Exponents Solution:

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Evaluate. Slide 5.1- 12 CLASSROOM EXAMPLE 4 Using Negative Exponents Solution:

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Special Rules for Negative Exponents If a ≠ 0 and b ≠ 0, then and Slide 5.1- 13 Define 0 and negative exponents.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the quotient rule for exponents. Objective 3 Slide 5.1- 14

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Quotient Rule for Exponents If a is any nonzero real number and m and n are integers, then That is, when dividing powers of like bases, keep the same base and subtract the exponent of the denominator from the exponent of the numerator. Slide 5.1- 15 Use the quotient rule for exponents. Be careful when working with quotients that involve negative exponents in the denominator. Write the numerator exponent, then a subtraction symbol, and then the denominator exponent. Use parentheses.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Apply the quotient rule, if possible, and write each result with only positive exponents. Cannot be simplified because the bases x and y are different. The quotient rule does not apply. Slide 5.1- 16 CLASSROOM EXAMPLE 5 Using the Quotient Rule for Exponents Solution:

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