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Solve Equations Using Nth Roots

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  1. 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.

  2. 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.

  3. 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

  4. 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.

  5. x3 = 64 = 3 64 x x = 4 for Examples 4 and 5 GUIDED PRACTICE Solve the equation. Round the result to two decimal places when appropriate. 13.x3 = 64 SOLUTION take 3rd root of each side. Simplify.

  6. 14. x5 = 512 1 1 x5 = 512 2 2 x5 = 1024 x = 5 1024 x = 4 for Examples 4 and 5 GUIDED PRACTICE SOLUTION Multiply each side by 2. take5throot of each side. Simplify.

  7. 15. 3x2 = 108 3x2 = 108 x2 = 36 x = + 6 x = 2 36 – for Examples 4 and 5 GUIDED PRACTICE SOLUTION Divide each side by3. take2ndroot of each side. Simplify.

  8. 16. x3 x3 = 2 = 2 x3 = 8 x = 3 8 x = 2 1 1 4 4 for Examples 4 and 5 GUIDED PRACTICE SOLUTION Multiply each side by4. take 3rd root of each side. Simplify.

  9. 17. ( x – 2 )3 = –14 ( x – 2 )3 = –14 ( x – 2 ) = 3 –14 x = 3 –14 + 2 x = 3 –14 + 2 x = – 0.41 for Examples 4 and 5 GUIDED PRACTICE SOLUTION take 3rd root of each side. add 2 to both sides. Write solution. Use a calculator.

  10. 18. ( x + 5 )4 = 16 ( x + 5 )4 = 16 ( x + 5 ) = +4 16 x = + 4 16 – 5 or x = 2 – 5 x = – 2 – 5 x or x = – 3 = –7 for Examples 4 and 5 GUIDED PRACTICE SOLUTION take 4th root of each side. add 5 to each side. Write solutions separately. Use a calculator.

  11. WHAT IF? Use the information from Example 5 to estimate the length of a coral cod that has the given weight. 19. w = 0.0167l 3 275 = 0.0167l 3 16467 = l 3 3 16467 = l 25 = l for Examples 4 and 5 GUIDED PRACTICE a. 275grams SOLUTION Write model for weight. Substitute 275 for w. Divide each side by 0.0167. Take cube root of each side. Use a calculator.

  12. ANSWER A coral cod that has the 275grams is about 25 cm long. for Examples 4 and 5 GUIDED PRACTICE

  13. w = 0.0167l 3 = 0.0167l 3 340 20360 = l 3 3 20360 = l = l 27 A coral cod that has the 340grams is about 27 cm long. ANSWER for Examples 4 and 5 GUIDED PRACTICE b. 340grams SOLUTION Write model for weight. Substitute 340 for w. Divide each side by 0.0167 Take cube root of each side Use a calculator

  14. w = 0.0167l 3 450 = 0.0167l 3 26946 = l 3 = l 30 = l 3 26946 A coral cod that has the 450grams is about 30 cm long. ANSWER for Examples 4 and 5 GUIDED PRACTICE c. 450grams SOLUTION Write model for weight. Substitute 450 for w. Divide each side by 0.0167 Take cube root of each side Use a calculator

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