Laws of radicals — lesson. Mathematics State Board, Class 9.

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Let us remember the formulas for solving and simplifying the radicals of positive integers $m$, $n$ and positive rational numbers $a$ and $b$.

The laws of radicals are:

S. No	Radical notation	Index notation
$1$	${(\sqrt[n]{a})}^{n} = a$	${(a^{\frac{1}{n}})}^{n} = a$
$2$	$\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab}$	$a^{\frac{1}{n}} \times b^{\frac{1}{n}} = {(ab)}^{\frac{1}{n}}$
$3$	$\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a} = \sqrt[n]{\sqrt[m]{a}}$	${(a^{\frac{1}{n}})}^{\frac{1}{m}} = a^{\frac{1}{mn}} = {(a^{\frac{1}{m}})}^{\frac{1}{n}}$
$4$	$\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$	$\frac{a^{\frac{1}{n}}}{b^{\frac{1}{n}}} = {(\frac{a}{b})}^{\frac{1}{n}}$

Example:

Write the surd $\sqrt{96}$ in its simplest form.

Solution:

$\sqrt{96} = \sqrt{2 \times 2 \times 2 \times 2 \times 2 \times 3}$

$= \sqrt{2^2 \times 2^2 \times 2 \times 3}$

$= \sqrt{2^2} \times \sqrt{2^2} \times \sqrt{2} \times \sqrt{3}$ (Law of radicals ($2$))

$= 2 \times 2 \times \sqrt{2} \times \sqrt{3}$ (Law of radicals ($1$))

$= 4 \times \sqrt{2} \times \sqrt{3}$

Therefore, the simplest form is $4 \times \sqrt{2} \times \sqrt{3}$.

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5. Laws of radicals