Question

FIND THE ∛ OF 188 (USE √

Answer 1

9514 1404 393

Answer:

 ∛188 ≈ 5.72865

Step-by-step explanation:

Any scientific or graphing calculator or spreadsheet can tell you the cube root of 188.

  ∛188 ≈ 5.72865431598...

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You know that 5³ = 125 and 6³ = 216, so the root will lie between 5 and 6, closer to 6. As a first approximation, you can figure it is about ...

  x = ∛188 ≈ 5 + (188-5³)/(6³ -5³) = 5 + 63/89 ≈ 5.71

You can figure this much using a 4-function calculator.

A closer approximation (x') can be had using the iteration formula ...

  x' = (2x³ +188)/(3x²)

For x = 5.71, the value of x' is ...

  x' ≈ (2×5.71³ +188)/(3×5.71²) ≈ 560.3388/97.8123 ≈ 5.7287

This value is correct when the root is rounded to 4 decimal places. Another execution of the iteration formula using this value will give the root accurate to 9 decimal places.