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Copyright © 2012 Pearson Education, Inc. 7.2 Rational Numbers as Exponents ■ Rational Exponents ■ Negative Rational Exponents ■ Laws of Exponents ■ Simplifying Radical Expressions Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Rational Exponents Consider a1/2a1/2. If we still want to add exponents when multiplying, it must follow that a1/2a1/2 = a1/2 + 1/2, or a1. This suggests that a1/2 is a square root of a. Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. When a is nonnegative, n can be any natural number greater than 1. When a is negative, n must be odd. Note that the denominator of the exponent becomes the index and the base becomes the radicand. Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Example Write an equivalent expression using radical notation. Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Solution The denominator of the exponent becomes the index. The base becomes the radicand. Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Write an equivalent expression using exponential notation. Example Solution The index becomes the denominator of the exponent. The radicand becomes the base. Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Positive Rational Exponents For any natural numbers m and n (n ≠1) and any real number a for which exists, Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Example Write an equivalent expression using radical notation and simplify. Solution Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Write an equivalent expression using exponential notation. Example Solution Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Negative Rational Exponents For any rational number m/n and any nonzero real number a for which exists, Caution! A negative exponent does not indicate that the expression in which it appears is negative. Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Example Write an equivalent expression with positive exponents and simplify, if possible. Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Solution 8-2/3 is the reciprocal of 82/3. Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Laws of Exponents The same laws hold for rational exponents as for integer exponents. Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Laws of Exponents For any real numbers a and b and any rational exponents m and n for which am, an, and bm are defined: 1. 2. 3. 4. In multiplying, add exponents if the bases are the same. In dividing, subtract exponents if the bases are the same. (Assume To raise a power to a power, multiply the exponents. To raise a product to a power, raise each factor to the power and multiply. Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Example Use the laws of exponents to simplify. Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Solution Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Simplifying Radical Expressions To Simplify Radical Expressions 1. Convert radical expressions to exponential expressions. 2. Use arithmetic and the laws of exponents to simplify. 3. Convert back to radical notation as needed. Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Example Use rational exponents to simplify. Do not use exponents that are fractions in the final answer. Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc. Solution Convert to exponential notation Simplify the exponent and return to radical notation Copyright © 2012 Pearson Education, Inc.