# What are the differences of radicals and rational exponents?

Jun 28, 2015

Combined with integer exponentiation, you can express the same things using either notation:

${x}^{\frac{p}{q}} \equiv \sqrt[q]{{x}^{p}}$

$\sqrt[n]{x} \equiv {x}^{\frac{1}{n}}$

#### Explanation:

If you combine a radical with an integer exponent then you can express the same concept as a rational exponent.

${x}^{\frac{p}{q}} \equiv \sqrt[q]{{x}^{p}}$

An $n$th root can be expressed as a rational exponent:

$\sqrt[n]{x} \equiv {x}^{\frac{1}{n}}$

The differences are basically notational.

Note that this assumes that $x > 0$. If $x \le 0$ or is a complex number then these identities do not always hold.