Question

How do you find

`\sqrt[3]{219}`
​

Answer 1

use an appropriate calculator

Explanation:

You didn't ask the value; you only asked how to find the value. Here is an answer for that question.

__

If your calculator does not have a cube-root function, you can use a 1/3 power:

`\sqrt[3]{219}=219^(1/3)`

Alternatively, you can use logarithms:

`\sqrt[3]{219}=10^{\left((1)/(3)\log{219}\right)}`

__

The Google calculator recognizes the cube root( ) function as shown in the attachment.

_____

Since you know 6^3 = 216, you know the cube root of 219 is only a little more than 6. A couple of iterations of ...

x' = (2/3)x +73/x^2

can refine your first guess of x=6 to 8 decimal places of accuracy, where x' is the next "guess". One more iteration will give the cube root to the accuracy displayed by most calculators.

__

In the above iteration formula, the value 73 is 219/3. This same sort of iterative technique can be used for any cube root. It calculates the next guess as the weighted average: 2/3 of the previous guess and 1/3 of the number divided by the square of the guess. Once you're in the ballpark, it will double the number of accurate decimal digits on each iteration.