Question

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

Answer 1

Hint: 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.