7.1 nth roots and rational exponents
- 1. 7.1 nth Roots and Rational Exponents
- 2. What is an “nth Root?” Extends the concept of square roots. For example: ◦ A cube root of 8 is 2, since 23 = 8 ◦ A fourth root of 81 is 3, since 34 = 81 For integers n greater than 1, if bn = a then b is an nth root of a. Written where n is the index of the radical.
- 3. Rational Exponents nth roots can be written using rational exponents. For example: In general, for any integer n greater than 1.
- 4. Real nth Roots If n is odd: ◦ a has one real nth root If n is even: ◦ And a > 0, a has two real nth roots ◦ And a = 0, a has one nth root, 0 ◦ And a < 0, a has no real nth roots
- 5. Finding nth Roots Find the indicated real nth root(s) of a. Example: n = 3, a = -125 n is odd, so there is one real cube root: (- 5)3 = -125 We can write
- 6. Example: n = 4, a = 16 n is even, and a > 0, so 16 has two real 4th roots: 24 = 16 and (-2)4 = 16 We can write
- 7. Your Turn! Find the indicated real nth root(s) of a. n = 4, a = 625 n = 3, a = -27
- 8. Rational Exponents Not always of the form 1/n. The denominator of the exponent is the index of the radical.
- 9. Evaluating with Rational Exponents Evaluate: 93/2 32-2/5
- 10. Your Turn! Evaluate: 493/2 16-3/4
- 11. Approximating with a Calculator Rewrite in rational exponent notation, then use calculator. Example: Use ( ) around the exponent! Your Turn! Approximate
- 12. Solving Equations with nth Roots Get the power alone, then take nth roots of both sides. Examples: 2x4 = 162 (x – 2)3 = 10
- 13. Your Turn! Solve each equation. 5x4 = 80 (x – 1)3 = 32