﻿ Writing Radicals in Rational Exponent Form

# Writing Radicals in Rational Exponent Form

Want some practice with the other direction?

When serious work needs to be done with radicals, they are usually changed to a name that uses exponents, so that the exponent laws can be used.

Also, this new name for radicals allows them to be approximated on any calculator that has a power key.

Here are the rational exponent names for radicals:

$\sqrt{x} = x^{1/2}$

$\root 3\of{x} = x^{1/3}$

$\root 4\of{x} = x^{1/4}$

$\root 5\of{x} = x^{1/5}$

And so on!

Regardless of the name used, the normal restrictions apply. For example, $\,x^{1/2}\,$ is only defined for $\,x\ge 0\,.$

## Examples

Write in rational exponent form:

$\root 7\of {x} = x^{1/7}$
$\cssId{s19}{\sqrt{x^3}} \cssId{s20}{= (x^3)^{1/2}} \cssId{s21}{= x^{3/2}}$
$\displaystyle \cssId{s22}{\frac{1}{\sqrt{x}}} \cssId{s23}{= \frac{1}{x^{1/2}}} \cssId{s24}{= x^{-1/2}}$
$\displaystyle \cssId{s25}{\frac{3}{\root 5\of{x}}} \cssId{s26}{= \frac{3}{x^{1/5}}} \cssId{s27}{= 3x^{-1/5}}$

## Concept Practice

Write in rational exponent form: