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In mathematics, a radical is any number that includes the root sign (√). The number under the root sign is a square root if no superscript precedes the root sign, a cube root is a superscript 3 precedes it (3√), a fourth root if a 4 precedes it (4√) and so on. Many radicals cannot be simplified, so dividing by one requires special algebraic techniques. To make use of them, remember these algebraic equalities:

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

## Numerical Square Root in the Denominator

In general, an expression with a numerical square root in the denominator looks like this:

\frac{a}{\sqrt{b}}

To simplify this fraction, you rationalize the denominator by multiplying the entire fraction by √​b​/√​b​.

Because

\sqrt{b} × \sqrt{b} = \sqrt{b^2} = b

the expression becomes

\frac{a\sqrt{b}}{b}

Examples:

1. Rationalize the denominator of the fraction

\frac{5}{\sqrt{6}}

Solution:​ Multiply the fraction by √6/√6

\frac{5\sqrt{6}}{\sqrt{6}\sqrt{6}} \\ \,\\ \frac{5\sqrt{6}}{6} \text{ or } \frac{5}{6}× \sqrt{6}

2. Simplify the fraction

\frac{6\sqrt{32}}{3\sqrt{8}}

Solution:​ In this case, you can simplify by dividing the numbers outside the radical sign and those inside it in two separate operations:

\frac{6}{3} = 2 \\ \,\\ \frac{\sqrt{32}}{ \sqrt{8}} = \sqrt{4} = 2

The expression reduces to

2 × 2 = 4

## Dividing by Cube Roots

The same general procedure applies when the radical in the denominator is a cube, fourth or higher root. To rationalize a denominator with a cube root, you have to look for a number, that when multiplied by the number under the radical sign, produces a third power number that can be taken out. In general, rationalize the number

\frac{a}{\sqrt[3]{b}} \text{ by multiplying by } \frac{ \sqrt[3]{b^2}}{\sqrt[3]{b^2}}

Example:

1. Rationalize

\frac{5}{\sqrt[3]{5}}

Multiply numerator and denominator by 3√25.

\frac{5 ×\sqrt[3]{25}}{\sqrt[3]{5} ×\sqrt[3]{25}} \\ \,\\ = \frac{5\sqrt[3]{25}}{\sqrt[3]{125}} \\ \,\\ = \frac{5\sqrt[3]{25}}{5}

\sqrt[3]{25}

## Variables with Two Terms in the Denominator

When a radical in the denominator includes two terms, you can usually simplify it by multiplying by its conjugate. The conjugate includes the same two terms, but you reverse the sign between them For example, the conjugate of

x + y \text{ is } x - y

When you multiply these together, you get

x^2 - y^2

Example:

1. Rationalize the denominator of

\frac{4}{x + \sqrt{3}}

Solution: Multiply top and bottom by x − √3

\frac{4(x - \sqrt{3})}{(x + \sqrt{3})(x - \sqrt{3} )}

Simplify:

\frac{4x - 4\sqrt{3}}{x^2 - 3}