Question

The square root of $6 + 4\sqrt 2 $ is:

Answer 1

Hint: Here we will find the square root by equating the given equation with $\sqrt a + \sqrt b $ and simplifying it further.

Complete step-by-step answer:
Given, $6 + 4\sqrt 2 = 6 + 2\sqrt 8 $
Let,
$\sqrt a + \sqrt b = \sqrt {6 + 2\sqrt 8 } \to (1)$
Squaring on both sides
$
  {(\sqrt a + \sqrt b )^2} = {(\sqrt {6 + 2\sqrt 8 } )^2} \\
   \Rightarrow a + b + 2\sqrt {ab} = 6 + 2\sqrt 8 \\
$
On comparing
$ \Rightarrow a + b = 6 \to (2){\text{ & ab = 8}}$
As you know ${(a - b)^2} = {(a + b)^2} - 4ab$
$
   \Rightarrow {(a - b)^2} = 36 - 32 = 4 \\
   \Rightarrow a - b = 2 \to (3) \\
$
Add equation (2) and (3)
$ \Rightarrow 2a = 8 \Rightarrow a = 4$
From equation 2
$ \Rightarrow a + b = 6 \Rightarrow b = 6 - 4 = 2$
From equation 1
$
  \sqrt a + \sqrt b = \sqrt {6 + 2\sqrt 8 } \\
   \Rightarrow \sqrt {6 + 2\sqrt 8 } = \sqrt 4 + \sqrt 2 = 2 + \sqrt 2 \\
$
So this is your required square root.

Note: In this type of question always assume a given square root is equal to $\sqrt a + \sqrt b $ and then simplify it will lead you to the desired answer.