Question

How do you express `x^(4/3)` in simplest radical form?

Answer 1

You raise `x` to the `4^"th"` power, then take the cube root.

Explanation:

When dealing with fractional exponents, it's always useful to remember that the exponent can be written as a product of an integer and of a fraction that has the numerator equal to 1.

In general, this looks like this

`a/b = a * 1/b`

This is important when dealing with fractional exponents because an exponent that takes the form `1/b`, like in the above example, is equivalent to taking the `b^"th"` root.

`x^(1/b) = root(b)(x)`

Since, for any `x>0`, you have `(x^a)^b = x^(a * b)`, you can write

`x^(4/3) = x^(4 * 1/3) = (x^4)^(1/3) = color(green)(root(3)(x^4))`

SImply put, you need to take the cube root from `x` raised to the `4^"th"` power.

Of course, you can also write

`x^(4/3) = x^(1/3*4) = (x^(1/3))^4 = color(green)((root(3)(x))^4`