Question

Simplify \[\sqrt {392} .\]

Answer 1

Hint:For simplifying the terms containing square root we have to remove all perfect squares if any present inside the square root.So here we have to factorize the given term and represent it in terms of perfect squares and then simplify it.

Complete step by step answer:
Given, \[\sqrt {392} ......................................\left( i \right)\]
Now we have to change the given term\[\sqrt {392} \] and represent them in terms of perfect squares such that we can take one of the terms that are repetitive outside the square root and thereby simplify it.Now we have to factorize $392$.
$392 = 2 \times 2 \times 2 \times 7 \times 7...........................\left( {ii} \right)$
On observing (ii) we see that the factorization of $392$gives$2 \times 2 \times 2 \times 7 \times 7$.Now we need to find $\sqrt {392} $which is $\sqrt {2 \times 2 \times 2 \times 7 \times 7} $.

Now we know that the terms which are repetitive, one of the terms can be taken out of the square root.So here we can see that $2,7$ are repeating and are appearing twice inside the square root, such that we can take one of the numbers which is one $2$ and one $7$ outside the square root. Thereby after taking one $2$ and one $7$ outside the square root we have only one $2$ inside the square root which is to be kept untouched.
Therefore we can write:
\[
\sqrt {392} = \sqrt {2 \times 2 \times 2 \times 7 \times 7} \\
\Rightarrow\sqrt {392} = 2 \times 7 \times \sqrt 2 \\
\therefore\sqrt {392} = 14\sqrt 2 ............................\left( {iii} \right) \\
 \]
Therefore by simplifying \[\sqrt {392} \] we get \[14\sqrt 2 \].

Note:Radical expressions are algebraic expressions which have or contain radicals, and the best way to solve a square root is to remove all the perfect squares from inside the square root if any exists. Also questions similar can be solved in a similar manner which is to factorize the given number and then taking one of the two digits outside the square root if multiple numbers exist.