6.1 - The “Nth” Root

1. 6.1 - The “Nth” Root

2. 1. √121 1 ANSWER 25 Lesson 6.1, For use with pages 414-419 Evaluate without using a calculator. 11 ANSWER 2. 5–2

3. –7 ± 2√6 4.A ball has a surface area sof 1253 square inches. use the formula r = √s to find the radius rof the ball. 1 3.54 Lesson 6.1, For use with pages 414-419 Evaluate without using a calculator. 3. Solve (x + 7)2 = 24. Write answers with radicals in simplest form. ANSWER about 10 in. ANSWER

4. 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

5. 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

6. 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

7. Keystrokes Expression Display 9 1 5 7 3 4 12 3 8 7  c. ( 43= 73/4 EXAMPLE 3 Approximate roots with a calculator a. 91/5 1.551845574 b. 123/8 2.539176951 4.303517071

8. for Examples 1, 2 and 3 GUIDED PRACTICE Find the indicated real nth root(s) of a. 1. n = 4, a = 625 3. n = 3, a = –64. SOLUTION ±5 SOLUTION –4 2.n = 6, a = 64 4. n = 5, a = 243 SOLUTION ±2 SOLUTION 3

9. 1 3 for Examples 1, 2 and 3 GUIDED PRACTICE Evaluate expressions without using a calculator. 5. 45/2 7. 813/4 27 SOLUTION 32 SOLUTION 6. 9–1/2 8. 17/8 SOLUTION SOLUTION 1

10. Expression 10. 64 2/3 – 11. (4√ 16)5 12. (3√–30)2 for Examples 1, 2 and 3 GUIDED PRACTICE Evaluate the expression using a calculator. Round the result to two decimal places when appropriate. 9. 42/5 1.74 SOLUTION SOLUTION 0.06 SOLUTION 32 9.65 SOLUTION

11. 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.

12. 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.

13. 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

14. 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.

15. 17. 18. 15. ( x – 2 )3 = –14 3x2 = 108 ( x + 5 )4 = 16 14. x5 = 512 16. x3 = 2 1 2 x = + 6 – x = 4 x = 4 x or x = – 3 = –7 x = 2 x = –0.41 1 4 for Examples 4 and 5 GUIDED PRACTICE Solve the equation. Round the result to two decimal places when appropriate. 13.x3 = 64 SOLUTION SOLUTION SOLUTION SOLUTION SOLUTION SOLUTION

16. WHAT IF? Use the information from Example 5 to estimate the length of a coral cod that has the given weight. 19. for Examples 4 and 5 GUIDED PRACTICE a. 275grams SOLUTION 25 cm b. 340grams SOLUTION 27 cm c. 450grams SOLUTION 30 cm