1Cademy - Rationalizing the Denominator

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Simplified Square Root

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Rationalizing the Denominator

Rationalizing the denominator is the process of converting a fraction that has a radical in the denominator into an equivalent fraction whose denominator is an integer (a rational number). Because square roots of non-perfect-square numbers are irrational, a fraction like $\frac{1}{\sqrt{2}}$ has an irrational denominator. To rationalize it, multiply both the numerator and the denominator by the same square root — this uses the Equivalent Fractions Property to preserve the fraction's value while eliminating the radical from the denominator.

The technique relies on the property that $\left(\sqrt{a}\right)^2 = a$ : squaring an irrational square root produces a rational number. So multiplying $\sqrt{a}$ by itself in the denominator replaces the radical with the integer $a$ .

For example, to rationalize $\frac{1}{\sqrt{2}}$ , multiply numerator and denominator by $\sqrt{2}$ :

$\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

The denominator $\sqrt{2} \cdot \sqrt{2} = 2$ is now an integer. A fraction with a square root in the denominator is not considered simplified — rationalizing the denominator is required even when a calculator is available.

When the denominator contains a coefficient multiplied by a square root (such as $3\sqrt{6}$ ), multiply only by the radical part ( $\sqrt{6}$ ), not the entire denominator. Always simplify the radical in the denominator first, before rationalizing — this keeps the numbers smaller and the arithmetic easier.

Updated 2026-04-21

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