Rational Exponents

MATH 017

Intermediate Algebra

S. Rook

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Overview

• Section 7.2 in the textbook– Simplify rational exponents– Apply rational exponents to simplify rational

expressions

Simplify Rational Exponents

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Rational Exponents

• Thus far, we have only seen integer exponents– Ex: 53, x-5

• Possible to have rational (i.e. fractional) exponents– Ex: 81/3, y-3/4

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Rational Exponents vs Radical Notation

• Relationship between rational exponents and radical notation

where p is power and r is radical

Ex:

• Most calculators take only up to the third root– How would we evaluate

prrp xx

24441

221

4 625

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Rational Exponents vs Radical Notation (Example)

Ex 1: Use radical notation to write the following and simplify if possible

51204183121 ,16,64,16 xx

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Rational Exponents vs Radical Notation (Example)

Ex 2: Use radical notation to write the following and simplify if possible

232343423 ,8,8,4 x

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Negative Rational Exponents

• Write any negative rational exponents as positive rational exponents

• Apply radical notation and simplify if possible

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Rational Exponents vs Radical Notation (Example)

Ex 3: Use radical notation to write the following and simplify if possible. Leave NO negative exponents

533245 32,8,16

Simplify Rational Exponent Expressions

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Simplify Rational Exponent Expressions

• Exponent rules for integer exponents apply to rational exponents as well– Remember them?

• Product: xa ∙ xb = xa+b • Quotient: xa / xb = xa-b • Power: (xa)b = xab

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Simplify Rational Exponent Expressions (Example)

Ex 4: Use the properties of exponents to simplify – write with only positive exponents

2321 cc

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Simplify Rational Exponent Expressions (Example)

Ex 5: Use the properties of exponents to simplify – write with only positive exponents

43

47

r

r

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Simplify Rational Exponent Expressions (Example)

Ex 6: Use the properties of exponents to simplify – write with only positive exponents

ts

ts2

23132

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Simplify Rational Exponent Expressions (Example)

Ex 7: Use the properties of exponents to simplify – write with only positive exponents

34121 32 xxx

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Simplify Rational Exponent Expressions (Example)

Ex 8: Use rational exponents to simplify the following – leave the final answer in radical notation

6 545 2 , wwyy

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Summary

• After studying these slides, you should know how to do the following:– Understand rational exponents. This includes

being able to convert to radical notation and simplify

– Simplify expressions containing rational expressions using the exponent rules