The radical and the fractional exponent

Radicals extend the concept of a power: writing $\sqrt[n]{a}$ is equivalent to writing $a^{1/n}$. This equivalence lets us apply all the rules of powers to roots as well.

Definition — The radical

The radical $\sqrt[n]{a}$ is made up of:

• Index: $n$ (the little number at the top left)
• Argument (or radicand): $a$

The equivalence with the power with a fractional exponent holds: $a^{\frac{m}{n}} = \sqrt[n]{a^m}, \qquad a\ge 0.$

Example

2^{\frac{3}{2}} &= \sqrt{2^3} = \sqrt{8} = 2\sqrt{2} \\[4pt] 2^{-\frac{5}{3}} &= \frac{1}{\sqrt[3]{2^5}} = \frac{1}{\sqrt[3]{32}} \\[4pt] \sqrt[3]{2}\cdot 2^x &= 2 \implies 2^{\frac{1}{3}+x} = 2^1 \implies x = \frac{2}{3} \end{aligned}$$

Links

Topics: Radicals
Concepts: Fractional exponent · Root index · Radical · Radicand
Skills: Using formulae