Question

Find the square root of:
${\text{6 + }}\sqrt {12} - \sqrt {24} - \sqrt 8 ?$

Answer 1

Hint: Try to take out common terms in order to make calculation easier.

Given: ${\text{6 + }}\sqrt {12} - \sqrt {24} - \sqrt 8 $
Rewriting above equation as:
$
   \Rightarrow \sqrt {36} + \sqrt {12} - \sqrt {24} - \sqrt 8 {\text{ }}\left\{ {\because \sqrt {36} = 6} \right\} \\
   \Rightarrow \sqrt {36} - \sqrt {24} + \sqrt {12} - \sqrt 8 \\
$
Taking $\sqrt {12} $ common from first two terms and $\sqrt 4 $ from other two terms, we get
$ \Rightarrow \sqrt {12} \left( {\sqrt 3 - \sqrt 2 } \right) + \sqrt 4 \left( {\sqrt 3 - \sqrt 2 } \right)$
Now, taking out $\left( {\sqrt 3 - \sqrt 2 } \right)$ common from above equation, we get:
$
   \Rightarrow \left( {\sqrt {12} + \sqrt 4 } \right)\left( {\sqrt 3 - \sqrt 2 } \right) \\
   \Rightarrow \sqrt 4 \left( {\sqrt 3 + \sqrt 1 } \right)\left( {\sqrt 3 - \sqrt 2 } \right){\text{ }} \ldots \left( 1 \right) \\
 $
Now, we know the value of $\sqrt 3 $ is $1.732$ and of $\sqrt 2 $ is $1.414$.
Hence, putting these values in $\left( 1 \right)$, we get
$
  \therefore 2 \times \left( {1.732 + 1} \right)\left( {1.732 - 1.414} \right) \\
   \Rightarrow 2 \times 2.732 \times 0.318 \\
   \Rightarrow 1.7375 \\
$

Note- Whenever there is a bigger term inside the square root whose value is very difficult to calculate, always try to decompose the bigger term into multiple smaller terms whose values are known to us.