Question

How do you rationalise the denominator and simplify $ \sqrt {\dfrac{{10}}{7}} $

Answer 1

Hint: The type of numbers that are divided into two parts by a horizontal line such that one number is present on the upper side of the horizontal line and one below it are called fractions, for example $ \dfrac{5}{3} $ is fraction. The upper part of the fraction is called the numerator while the lower part of the fraction is called the denominator. When the denominator of a fraction contains a radical/irrational number, we rationalize the fraction to make the fraction simpler and understandable, for rationalizing a fraction, we multiply both the numerator and the denominator of the fraction with the denominator but with opposite sign.

Complete step-by-step answer:
In the given question, the denominator is $ \sqrt 7 $ thus we rationalize this fraction, by multiplying both the numerator and denominator with $ - \sqrt 7 $ .
We rationalize the fraction $ \sqrt {\dfrac{{10}}{7}} $ as follows –
 $
  \sqrt {\dfrac{{10}}{7}} = \dfrac{{\sqrt {10} }}{{\sqrt 7 }} \\
   \Rightarrow \dfrac{{\sqrt {10} }}{{\sqrt 7 }} \times \dfrac{{ - \sqrt 7 }}{{ - \sqrt 7 }} \\
   \Rightarrow \dfrac{{ - \sqrt {70} }}{{ - 7}} \\
   \Rightarrow \dfrac{{\sqrt {70} }}{7} \;
  $
Hence the rationalized form of $ \sqrt {\dfrac{{10}}{7}} $ is $ \dfrac{{\sqrt {70} }}{7} $ .
On factorizing 70, we see that none of its factors is perfect square so it stays as it is and thus cannot be simplified further.
So, the correct answer is “ $ \dfrac{{\sqrt {70} }}{7} $”.

Note: By simplifying a fraction, we mean to express the fraction as a ratio of prime numbers or we can say that both the numerator and denominator should be prime numbers, that is, they should be divisible only by 1 and itself. For simplifying a fraction, we write it as a product of prime factors and then divide both of them with the common factors, in this question both the numerator and denominator are already prime numbers and thus the fraction cannot be simplified further.