Question

Find the rationalizing factor of $a\sqrt {ab}$ .

Answer 1

Rationalizing is the process of multiplying a surd with another surd to get a rational number. The elements or factors involved in the process of rationalization are called rationalization factors. Here we have been given $a\sqrt {ab}$ whose rationalizing factor is to be found. We also know that if two surds are multiplied and the corresponding result is a rational number then accordingly each element or the factor is the rationalizing factor of the other.

Complete answer:
Given the number $a\sqrt {ab}$
We know in order to find the rationalizing factor of $a\sqrt {ab}$ we need to convert the irrational number $a\sqrt {ab}$ into a rational number.
Hence we are multiplying $a\sqrt {ab}$ with $\sqrt {ab}$ so that we can convert $a\sqrt {ab}$ to a rational number. Since $\sqrt {ab}$ neutralizes the root of $a\sqrt {ab}$ and thereby makes it a rational number, we can write
$\Rightarrow \left( {a\sqrt {ab} } \right) \times \;\left( {\sqrt {ab} } \right)\; = a\left( {ab} \right)\;$
$\Rightarrow a\left( {ab} \right)\; = {a^2}b\;$
Here ${a^2}b$ is a rational number. So according to the definition $\sqrt {ab}$ would be the rationalizing factor of $a\sqrt {ab}$ .
Therefore the rationalizing factor of $a\sqrt {ab}$ is $\sqrt {ab}$ .

Note: In order to rationalize an irrational number with square root we have to multiply it with a number such that it neutralizes the square root and the square root goes away. Rationalizing factor is a term or a factor which is multiplied or divided to make the whole term rational. Also by multiplying two surds and if the product of the surds is a rational number then each of the surd is called a rationalizing factor. These points help in solving questions which are similar to the above mentioned question.