To multiply radical expressions, one can use the quotient rule for radicals. Let $u$ and $v$ be real numbers, variables or algebraic expressions. If the nth roots of $u$ and $v$ are real, the following rule is true.

$\sqrt[n]{\frac{u}{v}}$ = $\frac{\sqrt[n]{u}}{\sqrt[n]{v}}$, $v \neq 0$

To divide radicals one must use the quotient property of square roots. This property states that the square root of a quotient equals the quotient of the square roots.

$\sqrt{\frac{a}{b}}$ = $\frac{\sqrt{a}}{\sqrt{b}}$

where $x$ is a real number which is greater than or equal to $0$ and $y$ is real number that is greater than zero..

If one radical is a factor of the other, divide the radicals. If the factors have the a common factor, simplify.

Rationalizing the denominator: Occasionally we wish to express a fraction containing radicals in an equivalent form that containing no radicals in the denominator. This is accomplished by multiplying the numerator and the denominator by the expression that will remove the radical from the denominator. This process is called rationalizing the denominator.

1. If the denominator of a fraction is a square root, rationalize the denominator by multiplying the numerator and denominator by same appropriate square root that makes a perfect square radicand in the denominator.

2. If the denominator of a fraction is a cube root, rationalize the denominator by multiplying the numerator and denominator by the cube root that makes a perfect cube radicand in the denominator.

3. If the denominator of a fraction contains two terms with square roots, multiply the numerator and denominator by the conjugate of the denominator.

Examples:

1. $\frac{x}{\sqrt{5}}$ = $\frac{x\sqrt{5}}{\sqrt{5}\sqrt{5}}$ = $\frac{x\sqrt{5}}{5}$

2. $\frac{2}{\sqrt[3]{25}}$ = $\frac{2}{\sqrt[3]{25}}$ . $\frac{\sqrt[3]{5}}{\sqrt[3]{5}}$ = $\frac{2\sqrt[3]{5}}{5}$

3. $\frac{x}{\sqrt{x}-2}$ = $\frac{x(\sqrt{x}+2)}{(\sqrt{x}-2)(\sqrt{x}+2)}$ = $\frac{x(\sqrt{x}+2)}{x-4}$

To divide two radicals with the same index, write the quotient of the two radicands underneath the radical sign and, if possible simplify.

Example:

$\frac{\sqrt{8x^{12}}}{\sqrt{2x^{2}}}$ = $\sqrt{\frac{8}{2}.\frac{x^{12}}{x^{2}}}$

= $\sqrt{4x^{10}}$

= $\sqrt{4}\sqrt{x^{5}x^{5}}$

= $2x^{5}$

### Solved Examples

Question 1: Perform each division and simplify

$\frac{6}{\sqrt{x}-2}$

Solution:

Multiply numerator and denominator by conjugate of denominator.

Conjugate of $\sqrt{x}-2$ is $\sqrt{x}+2$

$\frac{6}{\sqrt{x}-2}$  = $\frac{6}{\sqrt{x}-2}$. $\frac{\sqrt{x}+2}{\sqrt{x}+2}$

= $\frac{6(\sqrt{x}+2)}{(\sqrt{x})^{2}-2^{2}}$

= $\frac{6\sqrt{x}+12}{x-4}$

Question 2: Perform the division and simplify

$\frac{1}{\sqrt{x}-\sqrt{x+1}}$

Solution:

Multiply numerator and  denominator by conjugate of denominator.

$\frac{1}{\sqrt{x}-\sqrt{x+1}}$ = $\frac{1}{\sqrt{x}-\sqrt{x+1}}$.$\frac{\sqrt{x}+\sqrt{x+1}}{\sqrt{x}+\sqrt{x+1}}$

=  $\frac{\sqrt{x}+\sqrt{x+1}}{(\sqrt{x})^{2}-(\sqrt{x+1})^{2}}$

= $\frac{\sqrt{x}+\sqrt{x+1}}{x-(x+1)}$

= $\frac{\sqrt{x}+\sqrt{x+1}}{-1}$

= $-\sqrt{x}-\sqrt{x+1}$

The following are the examples of dividing radicals.

### Solved Examples

Question 1: Simplify the radical expression.

$\frac{\sqrt{16a^{3}x}}{\sqrt{2ax}}$

Solution:

$\frac{\sqrt{16a^{3}x}}{\sqrt{2ax}}$ = $\sqrt{\frac{16a^{3}x}{2ax}}$

= $\sqrt{8a^{2}}$ = $2a\sqrt{2}$

Question 2: Simplify the radical expression.

$\frac{3\sqrt {12}}{\sqrt{10}}$

Solution:

$\frac{3\sqrt {12}}{\sqrt{10}}$ = 3$\sqrt{\frac{12}{10}}$ = 3$\sqrt{\frac{6}{5}}$

= $\frac{3\sqrt {6}}{\sqrt{5}}$

= $\frac{3\sqrt {6}}{\sqrt{5}}$. $\frac{\sqrt {5}}{\sqrt{5}}$

= $\frac{3\sqrt {30}}{5}$

Question 3: Simplify $\frac{\sqrt[3]{32}}{\sqrt[3]{4}}$

Solution:

$\frac{\sqrt[3]{32}}{\sqrt[3]{4}}$  = $\sqrt[3]{\frac{32}{4}}$

= $\sqrt[3]{8}$ = $2$