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Section 7.1

Section 7.1. nth roots and Rational exponents. Evaluating n th Roots. If b n = a , then b is an n th root of a. Real n th Roots. Let n be an integer greater than 1 and let a be a real number. If n is odd, then a has one real n th root:

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Section 7.1

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  1. Section 7.1 nth roots and Rational exponents

  2. Evaluating nth Roots • If bn = a, then b is an nth root of a.

  3. Real nth Roots • Let n be an integer greater than 1 and let a be a real number. • If n is odd, then a has one real nth root: • If n is even and a > 0, then a has two real nth roots:

  4. Real nth Roots • If n is even and a = 0, then a has one real nth root: • If n is even and a < 0, then a has NO real nth roots.

  5. Finding nthRoots • Find the indicated real nthroot(s) of a. • n = 3, a = -125 = -5 • n = 4, a = 16 = 2 or -2

  6. Rational Exponents • Let a1/nbe an nth root of a, and let m be a positive integer.

  7. Evaluating Expressions with Rational Exponents = 27 = 1/4

  8. Solving Equations Using nth Roots

  9. nth Roots and Rational Exponents You know that 3 is the square root of 9 because 9 is “3 squared” Roots exist other than “square” roots. Consider the following... 2 is the cube root of 8 since 23 = 8 5 is a fourth root of 625 since 54 = 625 –5 is a fourth root of 625 since (–5)4 = 625 These roots can be written with two different types of notation…Radical Notation or Rational Exponent Notation These are both asking for the third (or cube) root of 8... …which is 2 since 23 = 8

  10. Examples: =4 Simplify: 625 –1/4 This is the same as 1 625 1/4 or Simplify: 641/3 This is the same as (43)1/3 =4 or641/3 = ± 1 5

  11. To simplify rational exponents, you may use the following: am/n = (a1/n)m Example: Simplify 82/3 = (81/3)2 = (2)2 = 4 Example: Simplify 64-2/3 = (641/3)-2 = (4)-2 = 1 (4)2 = 1 16

  12. Solving Equations Using nth Roots

  13. Assignment Section 7.1: page 404-405 # 15 – 60 (every 3rd), 65, 66

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