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7.1 nth Roots and Rational Exponents 3/1/2013

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n th Root Ex. 3 2 = 9, then 3 is the square root of 9. If b 2 = a, then b is the square root of a. If b 3 = a, then b is the cube root of a. If b 4 = a, then b is the fourth root of a. If b n = a, then b is the nth root of a. You can write the n th root of a as Where a is a real number and n is the index of the radical.

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

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Example 1 Find nth Root(s) Find the indicated nth root(s) of a. a. = n 3,3, = a64 – b. = n 4,4, = a81 SOLUTION a. Because n is odd, 64 has one real cube root. – b.Because n is even and a is greater than 0, 81 has two real fourth roots. CHECK ()3)3 4 – = () 4 – () 4 – () 4 – = 64 – 3 – = 4– 4 81 = 3 4 = 3 –– and

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Extra Practice Find the indicated nth root(s) of a. Find nth Roots and Solve Equations Using nth Roots 1. = n 2,2, = a144 ANSWER 12 – 12, 2. = n 3,3, = a1000 ANSWER 10 3. = n 4,4, = a256 ANSWER 4 – 4,4,

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Example 2 Solve Equations Using nth Roots Solve the equation. a. = 2x 42x 4 162 SOLUTION a. = 2x 42x 4 162 Write original equation. = 22 162 2x 42x 4 Divide each side by 2. = x 4x 4 81 Simplify. = x3 – + 4 = x 4x 4 4 81 – + Take fourth root of each side.

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Example 2 Solve Equations Using nth Roots b.

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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.

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

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

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Extra Practice Evaluate the expression. ANSWER 5 Evaluate Expressions 4. 25 1/2 5. 81 1/2 ANSWER 9 6. 125 1/3 ANSWER 5 7. 32 1/5 ANSWER 2

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

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

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Example 4 Rewrite Expressions () 4 3 5 =53/453/4 Rewrite using rational exponents. () 4 3 5 a. b.Rewrite using radicals. 72/572/5 72/572/5 = () 5 2 7 = 1 () 3 2 2 2 2/3 – – = 1 22/322/3 c.

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

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

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Example 5 Evaluate Expressions with Rational Exponents Evaluate the expression. a. 4 3/2 b. 8 2/3 – SOLUTION Use radicals to rewrite and evaluate each expression. a. 4 3/2 3 = () 4 = 2323 8 = b. 8 2/3 – = 1 = 1 () 3 2 8 = 12 = 4 1

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Checkpoint Evaluate the expression. 17. 25 3/2 18. 16 5/4 19. 8 5/3 – ANSWER 125 ANSWER 32 ANSWER 1 32 Rewrite and Evaluate Expressions with Rational Exponents

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

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