Laws of Radical Expressions

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.

LAW EXAMPLE Simplified Answer

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

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

(b)

\sqrt[3]{17} = 17^{\frac{1}{3}}

$\sqrt[n]{a b} = \sqrt[n]{a} • \sqrt[n]{b}$ (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}

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

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

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