Question

Find the square root of \[\sqrt{0.0676~\,\times \,0.04}\]. Choose the correct answer.
A. 52
B. 0.52
C. 0.052
D. 5.2

Answer 1

Hint: Find the square root of the individual number one by one and then multiply them to get the final answer. The square root of \[\sqrt{0.0676}=\text{0}\text{.26}\]and the square root of \[\sqrt{0.04}=\text{0}\text{.2}\]. The final number will be the multiple of these two numbers so obtained above.

Complete step-by-step answer:
In the question, we have to find the square root of the numbers formed by the \[0.0676~\,\times \,0.04\].
So here we will write the square root as \[\sqrt{0.0676~\,\times \,0.04}\]. So, next we will use the property of the exponents that \[\sqrt{a~\,\times \,b}=\sqrt{a}\times \sqrt{b}\]. Similarly, here we have a=0.0676 and b= 0.04. So applying the above formula, we get the square root of the given number \[\sqrt{0.0676~\,\times \,0.04}\], as follows:
\[\begin{align}
  & \Rightarrow \sqrt{0.0676~\,\times \,0.04} \\
 & \Rightarrow \sqrt{0.0676~\,}\times \sqrt{\,0.04} \\

\end{align}\]
Here, we will first find the square root of \[\sqrt{0.0676}\]. So, here we can rewrite it as:
\[\begin{align}
  & \Rightarrow \sqrt{0.0676}=\sqrt{\dfrac{676}{10000}} \\
 & \Rightarrow \sqrt{0.0676}=\sqrt{\dfrac{{{(26)}^{2}}}{{{(100)}^{2}}}}\,\,\,\,\,\,\,\,\,\,\because 676={{(26)}^{2}},10000={{(100)}^{2}} \\
 & \,\Rightarrow \sqrt{0.0676}={{\left( \dfrac{{{(26)}^{2}}}{{{(100)}^{2}}} \right)}^{\dfrac{1}{2}}}\,\,\,\,\,\,\,\because \sqrt{a}={{a}^{\dfrac{1}{2}}} \\
 & \Rightarrow \sqrt{0.0676}=\dfrac{(26)}{(100)} \\
 & \Rightarrow \sqrt{0.0676}=0.26 \\
\end{align}\]

Now, similarly we will find the square root of \[\sqrt{0.04}\], as shown below:
\[\begin{align}
  & \Rightarrow \sqrt{0.04}=\sqrt{\dfrac{4}{100}} \\
 & \Rightarrow \sqrt{0.04}=\sqrt{\dfrac{{{(2)}^{2}}}{{{(10)}^{2}}}}\,\,\,\,\,\,\,\,\,\,\because 4={{(2)}^{2}},100={{(10)}^{2}} \\
 & \,\Rightarrow \sqrt{0.04}={{\left( \dfrac{{{(2)}^{2}}}{{{(10)}^{2}}} \right)}^{\dfrac{1}{2}}}\,\,\,\,\,\,\,\because \sqrt{a}={{a}^{\dfrac{1}{2}}} \\
 & \Rightarrow \sqrt{0.04}=\dfrac{(2)}{(10)} \\
 & \Rightarrow \sqrt{0.04}=0.2 \\
\end{align}\]
So now, we have:
\[\begin{align}
  & \Rightarrow \sqrt{0.0676~\times 0.04} \\
 & \Rightarrow \sqrt{0.0676~}\times \sqrt{0.04} \\
 & \Rightarrow 0.26\times 0.2 \\
\end{align}\]

This is because the square root of \[\sqrt{0.0676}=\text{0}\text{.26}\]and the square root of \[\sqrt{0.04}=\text{0}\text{.2}\].
Now, we just have to find the number that is found by multiplication of the above numbers to get the required square root.
\[\begin{align}
  & \Rightarrow 0.26\,\times \,0.2 \\
 & \Rightarrow 0.052 \\
\end{align}\]
So here we will have three digits after the decimal. And also, this is the final answer of the square root of the given number \[0.0676~\,\times \,0.04\]
Hence the correct answer is option C, i.e, 0.052

Note: Here, there is another method to find the square root of the \[0.0676~\,\times \,0.04\]. So, in this method, we will find the number that is found after multiplication of the numbers \[0.0676~\,\times \,0.04=\text{0}\text{.002704}\]and then we will find the square roots as follows:
\[\Rightarrow \sqrt{0.002704}=\text{0}\text{.052}\]. So , this is the required square root of the given number.