Writing Radicals in Rational Exponent Form

Want some practice with the other direction? Writing Rational Exponents as Radicals

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:

$\root 7\of {x} = x^{1/7}$

$\cssId{s19}{\sqrt{x^3}} \cssId{s20}{= (x^3)^{1/2}} \cssId{s21}{= x^{3/2}}$

$\displaystyle \cssId{s22}{\frac{1}{\sqrt{x}}} \cssId{s23}{= \frac{1}{x^{1/2}}} \cssId{s24}{= x^{-1/2}}$

$\displaystyle \cssId{s25}{\frac{3}{\root 5\of{x}}} \cssId{s26}{= \frac{3}{x^{1/5}}} \cssId{s27}{= 3x^{-1/5}}$