The laws for radicals are derived directly from the laws for exponents by using the definition $\sqrt[n]{{a}^{m}}={a}^{\frac{m}{n}}$. The laws are designed to make simplification much easier.

• ${\left(\sqrt[n]{a}\right)}^{n}=a$ ${\left(\sqrt[3]{8}\right)}^{3}={\left(2\right)}^{3}=8$

• $\sqrt[n]{a}={a}^{\frac{1}{n}}$ (a) $\sqrt{5}={5}^{\frac{1}{2}}$

(b) $\sqrt[3]{17}={17}^{\frac{1}{3}}$

• $\sqrt[n]{ab}=\sqrt[n]{a}•\sqrt[n]{b}$ (a) $\sqrt[]{4•3}=\sqrt{4}•\sqrt{3}$ = 2$\sqrt{3}$

Write the radicand as a product (b) $\sqrt[3]{16}=\sqrt[3]{8•2}=\sqrt[3]{8}•\sqrt[3]{2}$    = $2\sqrt[3]{2}$

• $\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$     (a) $\sqrt[4]{\frac{16}{81}}=\frac{\sqrt[4]{16}}{\sqrt[4]{81}}$ = $\frac{2}{3}$

(b) $\sqrt{\frac{3}{16}}=\frac{\sqrt{3}}{\sqrt{16}}$   $\frac{\sqrt{3}}{4}$

• $\sqrt[n]{{a}^{m}}={\left(\sqrt[n]{a}\right)}^{m}$ $\sqrt[3]{{27}^{2}}={\left(\sqrt[3]{27}\right)}^{2}$ = (3)2 = 9

• $\sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}$ $\sqrt[2]{\sqrt[3]{256}}=\sqrt[2\cdot 3]{256}=\sqrt[6]{256}$   = 2

It is important to reduce a radical to its simplest form. Using the laws of radicals for multiplication, division, raising a power to a power, and taking the radical of a radical makes the simplification process for radicals much easier.