Question

Find the square root of \[11 + 2\sqrt {30} \].

Answer 1

Hint:
Square root of any number \[(x)\]is the number\[(y)\]such that \[{(y)^2} = x\].
For example, the square root of 9 is 3 and the square root of 4 is 2 as \[{(3)^2} = 9\]and \[{(2)^2} = 4\].
Every positive number \[(x)\] has two square roots: \[\left( {\sqrt x } \right)\], which is positive and \[ - \left( {\sqrt x } \right)\], which is negative.

Complete step by step solution:
Let, \[11 + 2\sqrt {30} = Y\]
Putting under root both sides, we get
\[
   \Rightarrow \sqrt Y = \sqrt {11 + 2\sqrt {30} } \\
   \Rightarrow \sqrt Y = \sqrt {5 + 6 + 2\sqrt {5 \times 6} } \\
   \Rightarrow \sqrt Y = \sqrt {{{\left( {\sqrt 5 } \right)}^2} + {{\left( {\sqrt 6 } \right)}^2} + 2\left( {\sqrt 5 } \right)\left( {\sqrt 6 } \right)} \\
   \Rightarrow \sqrt Y = {\sqrt {\left( {\sqrt 5 + \sqrt 6 } \right)} ^2} \\
   \Rightarrow \sqrt Y = {\left( {\sqrt 5 + \sqrt 6 } \right)^{\dfrac{1}{2} \times 2}} \\
   \Rightarrow \sqrt Y = \left( {\sqrt 5 + \sqrt 6 } \right) \\
\]

The square root of \[11 + 2\sqrt {30} \] is \[\left( {\sqrt 5 + \sqrt 6 } \right)\].

Note:
Whenever we have to find the square root of any number in the form \[a + \sqrt b \], try to split a into two parts such the question can be reduce to \[\left( {\sqrt c + \sqrt d } \right)\] form. Always have such a kind approach in solving such questions.