Section 1 Part 1 Chapter 5

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

Copyright © 2012, 2008, 2004 Pearson Education, Inc. We use exponents to write products of repeated factors. For example, 2 5 is defined as = 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

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

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 Use the product rule for exponents. Be careful not to multiply the bases. Keep the same base and add the exponents.

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 CLASSROOM EXAMPLE 1 Using the Product Rule for Exponents Solution:

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

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

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

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 Define 0 and negative exponents.

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

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

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

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

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 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.

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 CLASSROOM EXAMPLE 5 Using the Quotient Rule for Exponents Solution: