7.1 nth Roots and Rational Exponents

1. 7.1 nth Roots and Rational Exponents 3/1/2013

2. nth Root Ex. 32 = 9, then 3 is the square root of 9. If b2 = a, then b is the square root of a. If b3 = a, then b is the cube root of a. If b4 = a, then b is the fourth root of a. If bn = a, then b is the nth root of a. You can write the nth root of a as Where a is a real number and n is the index of the radical.

3. a Any real number Greater than 0 0 Less than 0 Number of Roots One Two One No Real Solution n Odd Even Number of Real Roots

4. Example 1 a. b. n 3, a 64 n 4, a 81 = = = = – SOLUTION a. Because n is odd, 64 has one real cube root. – = 3 – 64 – 4 CHECK ( )3 ( ( ( ) ) ) = = – – – – – 4 4 4 4 64 and = = – – 4 81 3 4 81 3 Find nth Root(s) Find the indicated nth root(s) of a. • Because n is even and a is greater than 0, 81 has two real fourth roots.

5. Extra Practice 12, 12 – ANSWER 2. n 3, a 1000 10 = = ANSWER 3. n 4, a 256 = = ANSWER 4, 4 – Find nth Roots and Solve Equations Using nth Roots Find the indicated nth root(s) of a. 1. n 2, a 144 = =

6. Example 2 a. 2x4 162 = SOLUTION a. 2x4 162 = Write original equation. 2x4 Divide each side by 2. = 162 2 2 x4 81 = Simplify. x4 = + Take fourth root of each side. 4 4 81 – x 3 = + Simplify. – Solve Equations Using nth Roots Solve the equation.

7. Example 2 Solve Equations Using nth Roots b. Divide both sides by -2 = Cube root both sides x = -5

8. When to have 1 answer instead of 2 answers when doing problems with When the problem says SOLVE, you may have 1 or 2 answers depending on the index. When the problem says EVALUATE, then you only have 1 answer.

9. Vocabulary Rational Exponents: exponents written as fractions Ex : Radical Form: In general:

10. 9 a. 91/2 3 = = b. 161/4 2 = = c. 641/3 4 = = 64 16 4 3 d. ( )1/4 – – 32 , no real solution 32 4 = Example 3 Evaluate Expressions with Rational Exponents Evaluate the expression.

11. Extra Practice 4. 5 ANSWER 251/2 9 ANSWER 5. 811/2 5 ANSWER 6. 1251/3 2 7. ANSWER 321/5 Evaluate Expressions Evaluate the expression.

12. Power and nth root Ex : Radical Form: In general: Note: denominator is the index of the radical and numerator is the exponent of the radical

13. Negative Rational Exponent negative exponent still “moves” power Ex : Radical Form: In general:

14. ( ) 3 a. Rewrite using rational exponents. 4 5 1 1 ( ) 3 4 5 = 53/4 2 2/3 = = – ( ) 2 22/3 3 2 b. Rewrite using radicals. 72/5 ( ) 2 5 7 = Example 4 Rewrite Expressions 72/5 c. 2 2/3 Rewrite using radicals. –

15. Rewrite and Evaluate Expressions with Rational Exponents Extra Practice ANSWER 22/5 ANSWER 9. 1 13 1/4 – 13 4 Rewrite the expression using rational exponents. ( ) 8. 2 5 2

16. Rewrite and Evaluate Expressions with Rational Exponents ( ) 2 ANSWER 15 3 1 11. ANSWER 11 1/3 – 11 3 1 12. ANSWER 29 2/5 – ( ) 2 29 5 Extra Practice Rewrite the expression using radicals. 10. 152/3

17. Example 5 b. 8 2/3 – SOLUTION Use radicals to rewrite and evaluate each expression. ( ) a. 3 43/2 23 8 = = = 4 1 1 1 1 b. 8 2/3 = = = = – ( ) 4 82/3 22 2 3 8 Evaluate Expressions with Rational Exponents Evaluate the expression. a. 43/2

18. Rewrite and Evaluate Expressions with Rational Exponents Checkpoint 17. ANSWER 125 253/2 18. ANSWER 165/4 32 1 19. ANSWER 32 8 5/3 – Evaluate the expression.

19. Homework: Prac A WS 7-9, 13-48