How to Divide With Radicals?

When the Radicals are present in Fractions and especially in the denominator, they are required to be rationalized whose procedure is very similar to rationalizing surds and complex Numbers. If a surd is supposed to be present in the fraction, then for we rationalize it by multiplying and dividing the fraction with its conjugate i.e. (a + √b) when surd is (a - √b) and vice – versa. For radicals we just have the root part and so the Rationalization is done by multiplying and dividing the fraction by that radical only. Radicals are generally of the form √b. Let us learn how to divide with radicals.

Suppose the fraction that we are going to consider has radical only in the denominator as: 24 /√20. First we check whether the radical is in its proper form or not. √20 is not in the proper form, so we factorize it to get the proper radical as: 4 √5. Our fraction becomes: 24 / 4√5 or 6 /√5.

Now we do rationalization of the radical to get a real denominator as follows: 6 /√5 = (6 /√5) * (√5 /√5) =6 √5 / (√5) ² = 6 √5 / 5.

In case denominator has radicals in the form of surds, then rationalization is done as: Suppose we have the fraction as: (4) / (4 - √5). Multiplying and dividing the fraction by the conjugate of surd i.e. (4 + √5).
(4) / (4 - √5) = (4) / (4 - √5) * (4 + √5) / (4 + √5) = 4 (4 + √5) / 4² - (√5) ² = (16 + (4 √5)) / 16 – 5
= (16 + (4 √5)) / 11

Thus denominator we get after rationalization is real and therefore division becomes comparatively easy.