Question

What is the square root of 3 divided by the square root of 12?

Answer 1

Hint: We are given the division of two numbers under the square root. We can take the square root for the whole number and then simplify the terms inside the root. Finally taking the square root for the remaining number inside the square root we obtain the final answer.

Complete step by step solution:
In the question, we have been asked to find the square root of 3 divided by the square root of 12. We can write this as,$\dfrac{\sqrt{3}}{\sqrt{12}}$ .
By the property of square root, we know that, if two numbers under square root are divided separately, we can take the square root for the whole number. That is, if a and b are two positive integers, then,
$\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}}$
Therefore, we can write $\dfrac{\sqrt{3}}{\sqrt{12}}$ as,
$\dfrac{\sqrt{3}}{\sqrt{12}}=\sqrt{\dfrac{3}{12}}$
Now, we need to simplify the term under the square root and get it into the simplest possible form. To do this we need to find common factors of 3 and 12. We can write them as,
$\begin{align}
  & \,\,\,3=3\times 1 \\
 & 12=3\times 4 \\
\end{align}$
So, we can cancel out 3 from both the numerator and denominator as shown,
$\begin{align}
  & \sqrt{\dfrac{3}{12}}=\sqrt{\dfrac{3\times 1}{3\times 4}} \\
 & \sqrt{\dfrac{3}{12}}=\sqrt{\dfrac{1}{4}} \\
\end{align}$
Finally, we can take the square root of the number in the square root as follows,
$\begin{align}
  & \,\,\,\,\,\,\,\,\sqrt{\dfrac{1}{4}}=\sqrt{\dfrac{1}{{{2}^{2}}}} \\
 & \,\,\,\,\,\,\,\,\sqrt{\dfrac{1}{4}}=\dfrac{1}{\sqrt{{{2}^{2}}}} \\
 & \,\,\Rightarrow \sqrt{\dfrac{1}{4}}=\dfrac{1}{2} \\
 & \Rightarrow \sqrt{\dfrac{3}{12}}=\dfrac{1}{2} \\
\end{align}$
Therefore, we have found the square of 3 divided by the square root of 12 to be $\dfrac{1}{2}$.

Note: The square root of a number is not always just the positive part but also the negative part. For example, as we have found $\dfrac{1}{2}$to be the answer, even $-\dfrac{1}{2}$ can be a valid answer to this question as squaring both of these leads to the number inside the root.