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Nth Roots and Rational Exponents: Examples, Evaluations, and Problem Solving

This article provides examples, evaluations, and problem-solving techniques for finding nth roots and using rational exponents. It includes step-by-step solutions and calculator usage.

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Nth Roots and Rational Exponents: Examples, Evaluations, and Problem Solving

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  1. a. Because n = 3 is odd and a = –216 < 0, –216 has one real cube root. Because (–6)3= –216, you can write = 3√–216 = –6 or (–216)1/3 = –6. b. Because n = 4 is even and a = 81 > 0, 81 has two real fourth roots. Because 34 = 81 and (–3)4 = 81, you can write ±4√ 81 =±3 EXAMPLE 1 Find nth roots Find the indicated real nth root(s) of a. a. n = 3, a = –216 b. n = 4, a = 81 SOLUTION

  2. 1 1 23 323/5 64 ( )3 = (161/2)3 = 43 = 43 = 64 = = 16  1 1 1 1 1 1 = = = = ( )3 (321/5)3 323/5 32  5 8 23 8 = = = = EXAMPLE 2 Evaluate expressions with rational exponents Evaluate (a) 163/2 and (b)32–3/5. SOLUTION Radical Form Rational Exponent Form a. 163/2 163/2 b. 32–3/5 32–3/5

  3. Keystrokes Expression Display 9 1 5 7 3 4 12 3 8 7  c. ( 4 )3 = 73/4 EXAMPLE 3 Approximate roots with a calculator a. 91/5 1.551845574 b. 123/8 2.539176951 4.303517071

  4. a. 4x5 = 128 x5 = 32 x = 32  5 x 2 = EXAMPLE 4 Solve equations using nth roots Solve the equation. Divide each side by 4. Take fifth root of each side. Simplify.

  5. b. (x – 3)4 = 21 + x – 3 = 21 – 4  + x = 21 + 3 – 4  or 21 + 3 x = x = – 21 + 3 4  4  5.14 0.86 x or x EXAMPLE 4 Solve equations using nth roots Take fourth roots of each side. Add 3 to each side. Write solutions separately. Use a calculator.

  6. Biology A study determined that the weight w(in grams) of coral cod near Palawan Island, Philippines, can be approximated using the model w = 0.0167ℓ3 where l is the coral cod’s length (in centimeters). Estimate the length of a coral cod that weighs 200grams. EXAMPLE 5 Use nth roots in problem solving

  7. 0.0167 l 3 w = 0.0167 l 3 200 = 11,976 l 3 11,976 l  3 22.9 l A coral cod that weighs 200grams is about 23centimeters long. ANSWER EXAMPLE 5 Use nth roots in problem solving SOLUTION Write model for weight. Substitute 200 for w. Divide each side by 0.0167. Take cube root of each side. Use a calculator.

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