How to Rationalize the Denominator

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You can't solve an equation that contains a fraction with an irrational denominator, which means that the denominator contains a term with a radical sign. This includes square, cube and higher roots. Getting rid of the radical sign is called rationalizing the denominator. When the denominator has one term, you can do this by multiplying the top and bottom terms by the radical. When the denominator has two terms, the procedure is a little more complicated. You multiply the top and bottom by the conjugate of the denominator and expand and simply the numerator.

To rationalize a fraction, you have to multiply the numerator and denominator by a number or expression that gets rid of the radical signs in the denominator.

Rationalizing a Fraction with One Term in the Denominator

A fraction with the square root of a single term in the denominator is the easiest to rationalize. In general, the fraction takes the form ​a​ / √​x​. You rationalize it by multiplying the numerator and denominator by √​x​.

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

Since all you've done is multiply the fraction by 1, its value hasn't changed.

Example:

Rationalize

\frac{12}{\sqrt{6}}

Multiply the numerator and denominator by √6 to get

\frac{12\sqrt{6}}{6}

You can simplify this by dividing 6 into 12 to get 2, so the simplified form of the rationalized fraction is

2\sqrt{6}

Rationalizing a Fraction with Two Terms in the Denominator

Suppose you have a fraction in the form

\frac{a + b}{\sqrt{x} + \sqrt{y}}

You can get rid of the radical sign in the denominator by multiplying the expression by its conjugate. For a general binomial of the form ​x​ + ​y​, the conjugate is ​x​ − ​y​. When you multiply these together, you get ​x2 − ​y2. Applying this technique to the generalized fraction above:

\frac{a + b}{\sqrt{x} + \sqrt{y}} × \frac{\sqrt{x} - \sqrt{y}}{\sqrt{x} - \sqrt{y}} \\ \,\\ (a + b) × \frac{\sqrt{x} - \sqrt{y}}{x - y}

Expand the numerator to get

\frac{a\sqrt{x} -a\sqrt{y} + b\sqrt{x} - b\sqrt{y}}{x - y}

This expression becomes less complicated when you substitute integers for some or all of the variables.

Example:

Rationalize the denominator of the fraction

\frac{3}{1 - \sqrt{y}}

The conjugate of the denominator is 1 − ( −√​y​) = 1+ √​y​. Multiply the numerator and denominator by this expression and simplify:

\frac{3 × (1 + \sqrt{y})}{1 - y} \\ \,\\ \frac{3 + 3\sqrt{y}}{1 - y}

Rationalizing Cube Roots

When you have a cube root in the denominator, you have to multiply the numerator and denominator by the cube root of the square of the number under the radical sign to get rid of the radical sign in the denominator. In general, if you have a fraction in the form ​a​ / 3√​x​, multiply top and bottom by 3√​x2.

Example:

Rationalize the denominator:

\frac{7}{\sqrt[3]{x}}

Multiply the numerator and denominator by 3√​x2 to get

\frac{7 × \sqrt[3]{x^2} }{ \sqrt[3]{x} × \sqrt[3]{x^2} }= \frac{7 × \sqrt[3]{x^2} }{ \sqrt[3]{x^3}} \\ \,\\ \frac{7 \sqrt[3]{x^2}}{x}

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