Question

How do you rationalise the denominator and simplify $\dfrac{1}{{\sqrt 3 }}$ ?

Answer 1

In the given fraction, we have an irrational number in the denominator. To simplify the fraction, first we rationalise the denominator in the fraction so as to eliminate radical from the denominator. To get rid of the irrational number in the denominator, we multiply the numerator and denominator by $\sqrt 3$. Then by simplification, we get the required result.

Complete step by step explanation:
The given question in the form of $\dfrac{a}{b}$, where a is the numerator and b is the denominator. This form is known as fraction. There are mainly two types of fraction namely:
Proper fraction: If the numerator is smaller than the denominator is known as proper fraction.
Improper fraction: If the numerator is greater than or equal to the denominator of a fraction, then it is called an improper fraction. In such case, you could convert it into a whole number or mixed number fraction
Simplification of fraction means to reduce the given fraction into the simplest form.
Here in this question, we have to simplify the given fraction $\dfrac{1}{{\sqrt 3 }}$.
Generally in mathematics, we don’t desire to have an irrational number in the denominator. So, in order to get rid of the square root, we multiply the fraction by another fraction whose numerator and denominator are equal so as to keep the given fraction unchanged.
Considering the fraction given to us, $\dfrac{1}{{\sqrt 3 }}$. We multiply the numerator and denominator of the fraction by $\sqrt 3$ so as to get rid of the irrational number in the denominator.
Multiplying the fraction with another fraction whose both numerator and denominator are $\sqrt 3$. By doing this, we actually are not changing the value of the given fraction, since the value of $\dfrac{{\sqrt 3 }}{{\sqrt 3 }}$ equals $1$.
So, $\dfrac{1}{{\sqrt 3 }} \times \dfrac{{\sqrt 3 }}{{\sqrt 3 }}$
$\Rightarrow \dfrac{{\sqrt 3 }}{{{{\left( {\sqrt 3 } \right)}^2}}}$
Now, We know that $\sqrt a \times \sqrt a = {\left( {\sqrt a } \right)^2} = a$. So, we get,
$\Rightarrow \dfrac{{\sqrt 3 }}{3}$
Hence, the value of the given fraction $\dfrac{1}{{\sqrt 3 }}$ is $\dfrac{{\sqrt 3 }}{3}$.

Note: If the denominator of a fraction has square root, we rationalise the rational number. We multiply the numerator and denominator by the same number so as not to change the value of the fraction and get rid of the irrational number in the denominator.