How to Rationalize Denominators with Radicals

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Fractions may contain square root expressions in the numerator and the denominator. If the numerator and denominator contain the same square root, then you can find an equivalent fraction that has no square roots in the denominator. This process is called rationalizing the denominator.

To rationalize the denominator, you must multiply the top and bottom of the fraction by the conjugate of the denominator. Conjugate pairs differ only in the +/- sign between the whole number part and the radical part. Some conjugat pairs are

3 + √7  and  3 - √7
-5 + √2  and  -5 - √2
13 - 4√5  and  13 + 4√5
10 + 2√22  and  10 - 2√22

You can follow the example below to learn how to simplify fractions this way, or use the calculator on the left.

Example

Rationalize the denominator of the fraction

56 + 27√13
——————————
29 - 6√13

The first step is to multiply the top and bottom by the conjugate of 29 - 6√13, which is 29 + 6√13.

56 + 27√13   29 + 6√13
—————————— x —————————
29 - 6√13    29 + 6√13

When you multiply the two numerators, the new numerator is

56x29 + 56x6x√13 + 29x27x√13 + 27x6x13
= 1624 + 336√13 + 783√13 + 2106
= 3730 + 1119√13

And when you multiply the two denominators, the new denominator is

29x29 + 29x6x√13 - 29x6x√13 - 6x6x13
= 841 - 468
= 373

Notice how the radical disappears when you multiply two conjugates. Now the equivalent fraction is

3730 + 1119√13
——————————————
373

Which simplifies to 10 + 3√13.