Question

How to transform an expression to radical form `(125)^(-1/3)`?

Answer 1

You use the fact that `x^(1/n)` is equivalent to `root(n)(x)`.

Explanation:

In order to convert your expression to radical form, you need to use two proprties of fractional exponents. The first one tells you that

`x^(-y) = 1/x^y`

In your case, the exponent is equal to `-1/3`, which means that the expression is equivalent to

`(125)^(-1/3) = 1/(125)^(1/3)`

Next, use the fact that raising a number to a fractional exponent that has the form `1/n` is equivalent to extracting the `n^"th"` root of that number.

In your case, the fractional exponent is `1/3`, which means that you have

`1/(125)^(1/3) = 1/root(3)(125)`

As it turns out, `125` is a perfect cube, which means that you can write it as

`125 = 25 * 5 = 5 * 5 * 5 = 5^3`

The expression becomes

`1/root(3)(125) = 1/root(3)(5^3) = color(green)(1/5)`