Question

Write the lowest rationalising factor of : $\sqrt {13} + 3$ is $\sqrt {13} - m$, then $m$ is

Answer 1

Hint: First of all, write the irrational number which is to rationalised then, find the corresponding irrational factor which when multiplied with $\sqrt {13} + 3$ will convert into the form $\left( {a + b} \right)\left( {a - b} \right)$. Then, use the rationalising factor given in the question to compare and find the value of $m$

Complete step by step Answer:

We rationalize any irrational number to convert it into a rational number.
We generally rationalise with a number such that when it is multiplied with the given number, it forms the formula $\left( {a + b} \right)\left( {a - b} \right)$ which is equal to \[{a^2} - {b^2}\]
During, rationalizing, the rationalizing factor is multiplied and divided so, that the value of the expression is not changed.
The number which we rationalize is also an irrational number.
We are given that the number is $\sqrt {13} + 3$, which is an irrational number.
In order to rationalise it, we will use the rationalising factor $\sqrt {13} - 3$ so that it forms the expression $\left( {\sqrt {13} + 3} \right)\left( {\sqrt {13} - 3} \right) = {\left( {\sqrt {13} } \right)^2} - {3^2} = 4$
But, we are given that the rationalising factor is $\sqrt {13} - m$.
On comparing $\sqrt {13} - m$ with $\sqrt {13} - 3$, we get,
$m = 3$
Therefore, the value of m is 3.
Note: Rationalisation is the method by which we convert an irrational number to a rational number. Many students make mistake by taking the rationalizing factor as the number equal to the given number, which is incorrect. We have to take the rationalising factor as the number such that the sign of one of the term is different to form the formula $\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$. Also, if the irrational number is of the type $\sqrt a $, then simplify multiply and divide by the same number to rationalize, that is the rationalizing factor of $\sqrt a $ is the number itself.