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Chapter 8

Chapter 8. Section 7. Using Rational Numbers as Exponents. Define and use expressions of the form a 1/ n . Define and use expressions of the form a m / n . Apply the rules for exponents using rational exponents. Use rational exponents to simplify radicals. 8.7. 2. 3. 4. Objective 1.

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Chapter 8

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  1. Chapter 8 Section 7

  2. Using Rational Numbers as Exponents Define and use expressions of the form a1/n. Define and use expressions of the form am/n. Apply the rules for exponents using rational exponents. Use rational exponents to simplify radicals. 8.7 2 3 4

  3. Objective 1 Define and use expressions of the form a1/n. Slide 8.7-3

  4. Now consider how an expression such as 51/2 should be defined, so that all the rules for exponents developed earlier still apply. We define 51/2 so that 51/2· 51/2 = 51/2 + 1/2 = 51 = 5. This agrees with the product rule for exponents from Section 5.1. By definition, Since both 51/2· 51/2andequal 5, this would seem to suggest that 51/2 should equal Similarly, then51/3should equal Define and use expressions of the form a1/n. Review the basic rules for exponents: Slide 8.7-4

  5. a1/n If a is a nonnegative number and n is a positive integer, then Define and use expressions of the form a1/n. Notice that the denominator of the rational exponent is the index of the radical. Slide 8.7-5

  6. Simplify. 491/2 10001/3 811/4 EXAMPLE 1 Using the Definition of a1/n Solution: Slide 8.7-6

  7. Objective 2 Define and use expressions of the form am/n. Slide 8.7-7

  8. Define and use expressions of the form am/n. Now we can define a more general exponential expression, such as 163/4. By the power rule, (am)n = amn, so However, 163/4 can also be written as Either way, the answer is the same. Taking the root first involves smaller numbers and is often easier. This example suggests the following definition fora m/n. am/n If a is a nonnegative number and m and n are integers with n > 0, then Slide 8.7-8

  9. Evaluate. 95/2 85/3 –272/3 EXAMPLE 2 Using the Definition of am/n Solution: Slide 8.7-9

  10. Earlier, a–n was defined as for nonzero numbers a and integers n. This same result applies to negative rational exponents. a−m/n If a is a positive number and m and n are integers, with n> 0, then Using the definition of am/n. A common mistake is to write 27–4/3 as –273/4. This is incorrect. The negative exponent does not indicate a negative number. Also, the negative exponent indicates to use the reciprocal of the base, not the reciprocal of the exponent. Slide 8.7-10

  11. Evaluate. 36–3/2 81–3/4 EXAMPLE 3 Using the Definition of a−m/n Solution: Slide 8.7-11

  12. Objective 3 Apply the rules for exponents using rational exponents. Slide 8.7-12

  13. All the rules for exponents given earlier still hold when the exponents are fractions. Apply the rules for exponents using rational exponents. Slide 8.7-13

  14. Simplify. Write each answer in exponential form with only positive exponents. EXAMPLE 4 Using the Rules for Exponents with Fractional Exponents Solution: Slide 8.7-14

  15. Simplify. Write each answer in exponential form with only positive exponents. Assume that all variables represent positive numbers. EXAMPLE 5 Using Fractional Exponents with Variables Solution: Slide 8.7-15

  16. Objective 4 Use rational exponents to simplify radicals. Slide 8.7-16

  17. Use rational exponents to simplify radicals. Sometimes it is easier to simplify a radical by first writing it in exponential form. Slide 8.7-17

  18. Simplify each radical by first writing it in exponential form. EXAMPLE 6 Simplifying Radicals by Using Rational Exponents Solution: Slide 8.7-18

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