2. Similarly, 8 following generalization: The denominator of the rational exponent is the index of the radical.

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1 RD. Rational Exponents Rational Exponents In sections P and RT, we reviewed properties of powers with natural and integral exponents. All of these properties hold for real exponents as well. In this section, we give meaning to expressions with rational exponents, such as aa, 8, or (xx) 0., and use the rational exponent notation as an alternative way to write and simplify radical expressions. Observe that 9 = = = 9. Similarly, 8 following generalization: For any real number aa and a natural number >, we have aa = aa. = = = 8. This suggests the Notice: The denominator of the rational exponent is the index of the radical. Caution! If aa < 0 and is an even natural number, then aa is not a real number. Converting Radical Notation to Rational Exponent Notation Convert each radical to a power with a rational exponent and simplify, if possible. Assume that all variables represent positive real numbers. a. b. 7xx c. bb a. = = ( ) = = Observation: simplification. Expressing numbers as powers of prime numbers often allows for further b. 7xx = (7xx ) = 7 (xx ) = ( ) xx = distribution of exponents change into a power of a prime number Note: The above example can also be done as follows: 7xx = xx = ( xx ) = c. 9 bb = 9 bb = (bb ) = bb, as bb > 0.

2 7 Observation: aa = aa = aa. Generally, for any real number aa 0, natural number >, and integral number mm, we have aa mm = (aa mm ) = aa mm Rational exponents are introduced in such a way that they automatically agree with the rules of exponents, as listed in section RT. Furthermore, the rules of exponents hold not only for rational but also for real exponents. Observe that following the rules of exponents and the commutativity of multiplication, we have aa mm = (aa mm ) = aa mm = aa mm, provided that aa exists. Converting Rational Exponent Notation to the Radical Notation Convert each power with a rational exponent to a radical and simplify, if possible. a. b. ( 7) c. xx a. = = b. ( 7) = 7 c. xx = xx = xx = Notice that 7 = 7 =, so ( 7) = 7. However, ( 9) 9, as ( 9) is not a real number while 9 = 9 =. Caution: A negative exponent indicates a reciprocal not a negative number! Also, the exponent refers to xx only, so remains in the numerator. Observation: If aa mm is a real number, then provided that aa 0. aa mm = mm aa,

3 8 Caution! Make sure to distinguish between a negative exponent and a negative result. Negative exponent leads to a reciprocal of the base. The result can be either positive or negative, depending on the sign of the base. For example, 8 = = 8, but ( 8) = ( 8) = = and 8 = 8 =. Applying Rules of Exponents When Working with Rational Exponents Simplify each expression. Write your answer with only positive exponents. Assume that all variables represent positive real numbers. a. aa aa b. c. xx 8 yy a. aa aa = aa + = aa 9 8 = aa b. = = = c. xx 8 yy = xx 8 yy = xx yy Evaluating Powers with Rational Exponents Evaluate each power. a. b. 8 a. = ( ) = = = b. 8 = = = It is helpful to change the base into a power of prime number, if possible. Observe that if mm in aa mm is a multiple of, that is if mm = kkkk for some integer kk, then aa kk = aa kk = aa kk Simplifying Radical Expressions by Converting to Rational Exponents Simplify. Assume that all variables represent positive real numbers. Leave your answer in simplified single radical form.

4 9 a. 0 b. xx xx c. a. 0 = ( 0 ) = = 8888 b. xx xx c. = xx xx = xx + = xx = xx xx = xx xx = = + = = = Another solution: divide at the exponential level This bracket is essential! add exponents as =+ t = = = + = = RT. Exercises Concept Check Match each expression from Column I with the equivalent expression from Column II.. Column I Column II. Column I Column II a. A. a. ( ) A. b. B. b. 7 B. c. C. c. C. 8 d. D. not a real number d. D. 9 e. ( ) E. e. E. not a real number f. F. f. ( ) F. Concept Check Write the base as a power of a prime number to evaluate each expression, if possible

5 0. ( ). ( ). 8. ( ) Concept Check Rewrite with rational exponents and simplify, if possible. Assume that all variables represent positive real numbers xx 8. yy 9. xx 0. xx yy. xx. aa Concept Check Rewrite without rational exponents, and simplify, if possible. Assume that all variables represent positive real numbers xx. aa 7 7. ( ) 8. ( ) 9. xx 0. xx yy Concept Check Use the laws of exponents to simplify. Write the answers with positive exponents. Assume that all variables represent positive real numbers.. 8. xx xx aa aa. 8. yy 7 7. xx 8 yy 8. aa bb 8 9. yy xx 0. aa bb. xx xx. xx xx Use rational exponents to simplify. Write the answer in radical notation if appropriate. Assume that all variables represent positive real numbers.. xx. aaaa. yy 8. xx yy yy 0. 8pp. (xx yy). (xx + ) 0. xx yy. aa 0 dd Use rational exponents to rewrite in a single radical expression in a simplified form. Assume that all variables represent positive real numbers aa aa 8. xx xx 9. xx xx 0. xxxx zz. xx. aa xx 8 aa

6 . 8xx. aa xx. xxxx. (xx) 7. xx xx xx 70. xx Discussion Point 7. Suppose someone claims that aa + bb expressions are equal: aa + bb = + 0 must equal aa + bb, since, when aa = and bb = 0, the two = = + 9 = aa + bb. Explain why this is faulty reasoning. Analytic Skills Solve each problem. 7. One octave on a piano contains keys (including both the black and white keys). The frequency of each successive key increases by a factor of. For example, middle C is two keys below the first D above it. Therefore, the frequency of this D is =. times the frequency of the middle C. one octave a. If two tones are one octave apart, how do their frequencies compare? C A C D b. The A tone below middle C has a frequency of 0 cycles per second. Middle C is keys above this A note. Estimate the frequency of middle C. 7. According to one model, an animal s heart rate varies according to its weight. The formula NN(ww) = 88ww gives an estimate for the average number NN of beats per minute for an animal that weighs ww pounds. Use the formula to estimate the heart rate for a horse that weighs 800 pounds. 7. Meteorologists can determine the duration of a storm by using the function defined by TT(DD) = 0.07DD, where DD is the diameter of the storm in miles and TT is the time in hours. Find the duration of a storm with a diameter of mi. Round your answer to the nearest tenth of an hour.