Question

How do you simplify $\sqrt{168}$ ?

Answer 1

Hint: Here we have to simplify the given value. Firstly as we can see that the value is given as a square root term so we need to remove that square root for that we will find the factor of the numbers inside the square root. Then we will make a pair of similar numbers and take them outside the square root values. Finally we will simplify further and get our desired answer.

Complete step-by-step solution:
We have to simplify the value given below:
$\sqrt{168}$…$\left( 1 \right)$
Firstly we will find the factor of $168$ using long division method where we divide the number $168$ by the natural numbers starting from $2$till we get the remainder as $1$ as follows:
$\begin{align}
  & 2\left| \!{\underline {\,
  168 \,}} \right. \\
 & 2\left| \!{\underline {\,
  84 \,}} \right. \\
 & 2\left| \!{\underline {\,
  42 \,}} \right. \\
 & 3\left| \!{\underline {\,
  21 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & \,\,\,\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
So we get the factor of $168$ as follows:
$168=2\times 2\times 2\times 3\times 7$
Put the above value in equation (1) as follows:
$\Rightarrow \sqrt{168}=\sqrt{2\times 2\times 2\times 3\times 7}$
Now make pair of similar number in above case we get only one pair as follows:
$\Rightarrow \sqrt{168}=\sqrt{\left( 2\times 2 \right)\times 2\times 3\times 7}$
We can take one among the two similar terms outside the square root as follows:
$\Rightarrow \sqrt{168}=2\sqrt{2\times 3\times 7}$
Multiply the term inside the square root,
$\Rightarrow \sqrt{168}=2\sqrt{42}$
So we get the answer as$2\sqrt{42}$ .
Hence on simplifying $\sqrt{168}$ we get the value $2\sqrt{42}$ .

Note: Factors are all the numbers that divide the given number completely. That is, the remainder is always zero. One is the common factor of all the numbers. Number of factors of every number is finite and each factor is always less than the number given. Square means when we multiply the number by itself and square root is an inverse operation of squaring a number. Every number has two square roots as the square of any number is always positive.