Question

Find the square root of: \[31 + 4\sqrt {21} \]
(A) $\sqrt {24} + \sqrt 3 $
(B) $\sqrt {28} + \sqrt 3 $
(C) $\sqrt {36} + \sqrt 3 $
(D) $\sqrt {28} + \sqrt 2 $

Answer 1

Hint: To find the square root of this type of expressions we have to suppose that their root is in this form of $\left( {\sqrt a + \sqrt b } \right)$ . Now, find the square of $\left( {\sqrt a + \sqrt b } \right)$ and equate the obtained value with the given expression that is $31 + 4\sqrt {21} $ and find the value of $a$ and $b$ then we get the required square root.

Complete step-by-step answer:
Here, the given expression is \[31 + 4\sqrt {21} \]
Suppose, the root of this expression is in the form of $\left( {\sqrt a + \sqrt b } \right)$
Now, we have to find the square of $\left( {\sqrt a + \sqrt b } \right)$
$ \Rightarrow {\left( {\sqrt a + \sqrt b } \right)^2} = {\left( {\sqrt a } \right)^2} + {\left( {\sqrt b } \right)^2} + 2\sqrt {ab} $
$ \Rightarrow {\left( {\sqrt a + \sqrt b } \right)^2} = a + b + \sqrt {ab} $
Since, we have supposed the root of the given expression in this form. so, we can equate the square of $\left( {\sqrt a + \sqrt b } \right)$ to the given expression.
$ \Rightarrow {\left( {\sqrt a + \sqrt b } \right)^2} = 31 + 4\sqrt {21} $
$ \Rightarrow a + b + 2\sqrt {ab} = 31 + 2\sqrt {4 \times 21} $
$ \Rightarrow a + b + 2\sqrt {ab} = 31 + 2\sqrt {84} $
By comparing the left hand side and right hand side of the equation we get,
$a + b = 31$ and $ab = 84$
Now, apply the formula ${\left( {x - y} \right)^2} = {\left( {x + y} \right)^2} - 4xy$ to find the value of $\left( {a - b} \right)$
$ \Rightarrow {\left( {a - b} \right)^2} = {\left( {31} \right)^2} - 4 \times 84$
$ \Rightarrow {\left( {a - b} \right)^2} = 961 - 336$
$ \Rightarrow {\left( {a - b} \right)^2} = 625$
$ \Rightarrow \left( {a - b} \right) = \sqrt {625} $
$\therefore \left( {a - b} \right) = 25$
We have two equations first is
$a + b = 31$ and second is $a - b = 25$
Now, adding these two equations we get,
$ \Rightarrow 2a = 56$
$ \Rightarrow a = \dfrac{{56}}{2} = 28$
$\therefore a = 28$
Putting the value of $a$in the equation $a + b = 31$ we get,
\[ \Rightarrow 28 + b = 31\]
$ \Rightarrow b = 31 - 28$
$\therefore b = 3$
Now, we have got the value of $a$ and $b$ so, the required root of the given expression is $\sqrt {28} + \sqrt 3 $

Thus, option ‘B’ is correct for the given question.

Note: The root of this type of expression can be found by hit and trial method.
Break the number $31$ (other than the number in root) in two parts such that the multiplication of the two numbers is equal to the number within root i.e $84$ in this question and then write the root as $\left( {\sqrt a + \sqrt b } \right)$ where $a$ and $b$ are the breakdown parts of number. In this problem $31 = 3 + 28$ where $a = 28$ and $b = 3$ and the multiplication of the $3$ and $28$ is $84$. Thus the required root of the given expression is $\sqrt {28} + \sqrt 3 $.