Question

If rationalization of $\dfrac{1}{{\sqrt {12} }}$ is $\dfrac{{\sqrt a }}{6}$, then value of a is___.

Answer 1

Hint- As, this is a question of rationalization, in this one we need to rationalize the denominator, it can be done by multiplying and dividing the expression by the real number present in the denominator, by doing the given expression will be rationalized.


Complete step by step answer:
The given irrational number is $\dfrac{1}{{\sqrt {12} }}$.
As we know that $\sqrt {12} \cdot \sqrt {12} $ is a rational number.
So, when $\dfrac{1}{{\sqrt {12} }}$ is multiplied and divided by $\sqrt {12} $, it becomes a rational number and also, the value of $\dfrac{{\sqrt {12} }}{{\sqrt {12} }}$ is 1.
So,
$\dfrac{1}{{\sqrt {12} }} \times \dfrac{{\sqrt {12} }}{{\sqrt {12} }} = \dfrac{{\sqrt {12} }}{{12}}$, this expression can be further simplified further,
As, $12 = 2 \times 2 \times 3$, use this in the above expression.
$\dfrac{{\sqrt {12} }}{{12}} = \dfrac{{\sqrt {2 \times 2 \times 3} }}{{12}} = \dfrac{{2\sqrt 3 }}{{12}} = \dfrac{{\sqrt 3 }}{6}$.
Comparing $\dfrac{{\sqrt a }}{6}$ with the evaluated value $\dfrac{{\sqrt 3 }}{6}$, then a value comes out to be 3.


Note- In this problem we have to rationalize the denominator.
To rationalize these types of problems we need to multiply and divide the expression by its denominator part.
If there is an expression in the denominator we need to multiply with conjugate to rationalize the denominator.
We generally rationalize the expression to get a real number that can be easily represented on the number line, as the irrational number is not easy to find or to represent on the number line.