Question

Find the square root of 0.09 using the long division method.

Answer 1

Hint: We solve this problem first converting the given number to fraction and then we find the square root using the long division method. For doing the long division we consider the numbers present as pairs from the right side and take a single number left as one pair. Then we use the long division from the left side taking the least square possible and continue the division until we get zero. Then the quotient will be the square root of the required number.

Complete step by step answer:
We are given that the number as 0.09
Let us assume that number as
\[{{x}^{2}}=0.09\]
Now, by converting the number to fraction we get
\[\begin{align}
  & \Rightarrow {{x}^{2}}=\dfrac{9}{100} \\
 & \Rightarrow x=\dfrac{\sqrt{9}}{\sqrt{100}}.......equation(i) \\
\end{align}\]
Now let us find the square root of the numerator that is 9.
By using the long division method we get
\[\begin{matrix}
   3 \\
   3\left| \!{\overline {\,
 \begin{align}
  & 9 \\
 & 9 \\
\end{align} \,}} \right. \\
   0 \\
\end{matrix}\]
Therefore, we can write
\[\Rightarrow 9={{3}^{2}}\]
So, we can say that the square root of 9 is 3.
Now, let us find the square root of 100 as follows
\[\begin{matrix}
   10 \\
   1\left| \!{\overline {\,
 \begin{align}
  & 100 \\
 & 1\downarrow \\
\end{align} \,}} \right. \\
   0\left| \!{\overline {\,
 \begin{align}
  & 000 \\
 & 0 \\
\end{align} \,}} \right. \\
\end{matrix}\]
Therefore, we can say that
\[\Rightarrow 100={{10}^{2}}\]
So, we can say that the square root of 100 is 10.
Now, by substituting the required values in the equation (i) we get
\[\begin{align}
  & \Rightarrow x=\dfrac{3}{10} \\
 & \Rightarrow x=0.3 \\
\end{align}\]

Therefore we can say that the square root of 0.09 is 0.3.

Note: This problem can be done in another way.
Here, rather than converting the number to fraction we can directly apply the long division method taking the digits in pairs from right side and continuing the division from left side.
In the number 0.09 we take 09 after decimal point as a pair and 0 before decimal point as one pair.
Now by applying the division we get
\[\begin{matrix}
   0.3 \\
   0\left| \!{\overline {\,
 \begin{align}
  & 0.09 \\
 & 0\downarrow \\
\end{align} \,}} \right. \\
   3\left| \!{\overline {\,
 \begin{align}
  & 009 \\
 & 009 \\
 & 0 \\
\end{align} \,}} \right. \\
\end{matrix}\]
Therefore the quotient is 0.3 so that the square root of 0.09 is 0.3.