Question

Find the square root of $7 - 2\sqrt {10} $
$
  (a){\text{ }}\sqrt {15} - \sqrt 2 \\
  (b){\text{ }}\sqrt 3 - \sqrt 2 \\
  (c){\text{ }}\sqrt 5 - \sqrt 3 \\
  (d){\text{ }}\sqrt 5 - \sqrt 2 \\
 $

Answer 1

Hint: In this question we have to find the square root of the given entity. So let the square root of the given entity be of the form \[\sqrt a - \sqrt b \], then square both the sides and apply necessary algebraic identities to get the answer.

Complete step-by-step answer:
Given equation

$7 - 2\sqrt {10} $

Square root of given equation is

\[\sqrt {7 - 2\sqrt {10} } \]

There are two terms in the given equation therefore in the square root of this it also has two terms.

So, let \[\sqrt a - \sqrt b = \sqrt {7 - 2\sqrt {10} } ................\left( 1 \right)\]

Squaring both sides

\[{\left( {\sqrt a - \sqrt b } \right)^2} = {\left( {\sqrt {7 - 2\sqrt {10} } } \right)^2}\]

Now, as we know that \[{\left( {a - c} \right)^2} = {a^2} + {c^2} - 2ac\] so use this property in above equation we have,

\[ \Rightarrow a + b - 2\sqrt {ab} = 7 - 2\sqrt {10} \]

So, on comparing

\[a + b = 7...................\left( 1 \right){\text{, }}\sqrt {ab} = \sqrt {10} {\text{ }}\]

So on squaring both sides we have,

\[{\left( {a + b} \right)^2} = {7^2}{\text{ = 49}}............\left( 2 \right){\text{, ab}} = 10.................\left( 3 \right)\]

Now it is a known fact that that ${\left( {a - b} \right)^2} = {\left( {a + b} \right)^2} - 4ab$
So from equation (2) and (3) we have,

${\left( {a - b} \right)^2} = 49 - 4\left( {10} \right) = 9$

Now take square root on both sides we have,

$ \Rightarrow \left( {a - b} \right) = \sqrt 9 = 3$…………………. (4)

From add equation (1) and (4) we have

$ \Rightarrow a + b + a - b = 7 + 3$

$ \Rightarrow 2a = 10$

$ \Rightarrow a = 5$

Now from equation (1)

$ \Rightarrow b = 7 - a = 7 - 5 = 2$

So the required square root is \[\sqrt 5 - \sqrt 2 \]

So, this is the required square root.

Hence option (d) is correct.

Note: Whenever we face such types of problems the key concept is simply to assume the square root of the given entity to be of a certain form. Good understanding of basic algebraic identities always helps in solving problems of this kind. These concepts will help in getting on the right track to reach the desired square root.