Simplify: $\dfrac{1}{\sqrt{2}}$

Hint:We first simplify the fraction keeping irrational values in the numerator. Then we use the division of square root process to find the square root of 2. We use the concept for both integer and decimal to find the value of $\sqrt{2}$ till the 3-digit place after decimal. Then we divide by 2.

Complete step by step answer:
We simplify $\dfrac{1}{\sqrt{2}}$ to keep irrational values in the numerator and get $\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}$
We first have to find and check if the number 2 is a square number or not. 2 is a prime number and that’s why it can’t be broken in factored form.We take 2 digits as a set from the right end and complete the division. For decimal form we take the set from the right side of the decimal.
\[\begin{align}
  & 1 \\
 & 1\left| \!{\overline {\,
 \begin{align}
  & \overline{2}.\overline{00}\overline{00} \\
 & \underline{1} \\
 & 1.000 \\
\end{align} \,}} \right. \\
\end{align}\]
Now we have to enter the decimal part. We keep doing the breaking in the set form till 3-digit place after decimal.
\[\begin{align}
  & 1.414 \\
 & 24\left| \!{\overline {\,
 \begin{align}
  & \overline{1}\overline{00}\overline{00}\overline{00} \\
 & \underline{96} \\
 & 400 \\
\end{align} \,}} \right. \\
 & 281\left| \!{\overline {\,
 \begin{align}
  & 400\overline{00} \\
 & \underline{281} \\
 & 11900 \\
\end{align} \,}} \right. \\
 & 2824\left| \!{\overline {\,
 \begin{align}
  & 11900 \\
 & \underline{11296} \\
 & 604 \\
\end{align} \,}} \right. \\
\end{align}\]
Now dividing by 2 we get, $\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}=\dfrac{1.414}{2}=0.707$.

Hence, the value of $\sqrt{2}$ is $1.414$.

Note: The long-division method and arranging the set of 2 digits is different for integer and decimal. But taking double for the next division and putting a particular number is the same process for both of them. Since 2 is a non-perfect square number, we will find the value of root 2 using the long division method as shown above.