Question

Find the square root of given :
$21 - \sqrt {440} $
$
  (a){\text{ }}\sqrt {12} - \sqrt {10} \\
  (b){\text{ }}\sqrt {11} - \sqrt 6 \\
  (c){\text{ }}\sqrt {11} - \sqrt {10} \\
  (d){\text{ }}\sqrt {10} - \sqrt 2 \\
 $

Answer 1

Hint – In this question let the square root of $21 - \sqrt {440} $ be of the form \[\sqrt a - \sqrt b \] that is \[\sqrt a - \sqrt b = \sqrt {21 - \sqrt {440} } \], then apply proper algebraic identities to get the value of a and b. Substitute them back to get the right answer.

Complete step-by-step answer:
Given equation
$21 - \sqrt {440} $
Square root of given equation is
\[\sqrt {21 - \sqrt {440} } \]
There are two terms in the given equation therefore in the square root of this it also has two terms.
So, let \[\sqrt a - \sqrt b = \sqrt {21 - \sqrt {440} } \]
Squaring both sides
\[{\left( {\sqrt a - \sqrt b } \right)^2} = {\left( {\sqrt {21 - \sqrt {440} } } \right)^2}\]
Now, as we know that \[{\left( {a - c} \right)^2} = {a^2} + {c^2} - 2ac\] so use this property in above equation we have,
\[ \Rightarrow a + b - 2\sqrt {ab} = 21 - \sqrt {440} \]
So, on comparing
\[a + b = 21...................\left( 1 \right){\text{, 2}}\sqrt {ab} = \sqrt {440} {\text{ }}\]
So on squaring both sides we have,
\[{\left( {a + b} \right)^2} = {\left( {21} \right)^2}{\text{ = 441}}............\left( 2 \right){\text{, 4ab}} = 440.................\left( 3 \right)\]
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,
${\left( {a - b} \right)^2} = 441 - 440 = 1$
Now take square root on both sides we have,
$ \Rightarrow \left( {a - b} \right) = \sqrt 1 = 1$…………………. (4)
From add equation (1) and (4) we have
$ \Rightarrow a + b + a - b = 21 + 1$
$ \Rightarrow 2a = 22$
$ \Rightarrow a = 11$
Now from equation (1)
$ \Rightarrow b = 21 - a = 21 - 11 = 10$
So the required square root is \[\sqrt {11} - \sqrt {10} \]
So, this is the required square root.
Hence option (C) is the correct answer.

Note – There is a specific format for solving problems of this kind. Which is to assume the square root in terms of some variables. Since the direct square root could only be found for numbers which are perfect square thus there is no direct way for finding the square root of numbers mentioned in this problem. The basic definition of square root is a number which produces a specified quantity when the number is multiplied to itself twice.