# Find the square root of:

${\text{6 + }}\sqrt {12} - \sqrt {24} - \sqrt 8 ?$

Answer

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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.

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.

Last updated date: 26th Sep 2023

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