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

Last updated date: 17th Mar 2023
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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.