Question

What is the square root of $\dfrac{48}{27}$?

Answer 1

Hint: Simplify the given expression by cancelling the common factors. To do this use the prime factorization method to write the given numerator and denominator as the product of their primes. Now, cancel the factors which are common in them. Now, try to write the remaining factors such that their exponent becomes 2. Take the square root and use the property of exponent ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ to get the answer.

Complete step by step solution:
Here we have been provided with the expression $\dfrac{48}{27}$ and we are asked to find its square root. Let us assume the given expression as E. So we have,
\[\Rightarrow E=\dfrac{48}{27}\]
Now, we have to check if there are common factors for the numerator and denominator or not. If there are then we have to cancel them and simplify the expression ‘E’. To do this, we will write the given numbers as the product of their primes. So we get,
\[\Rightarrow 48=2\times 2\times 2\times 2\times 3\] and \[\Rightarrow 27=3\times 3\times 3\]
Substituting this in expression E we get,
\[\Rightarrow E=\dfrac{2\times 2\times 2\times 2\times 3}{3\times 3\times 3}\]
Cancelling the common factors we get,
\[\Rightarrow E=\dfrac{2\times 2\times 2\times 2}{3\times 3}\]
Since we have to find the square root so we will try to write the factors in the exponential form such that its exponent becomes 2, so we have,
\[\Rightarrow E=\dfrac{{{2}^{2}}\times {{2}^{2}}}{{{3}^{2}}}\]
Now, taking square root both the sides we get,
\[\begin{align}
  & \Rightarrow \sqrt{E}=\sqrt{\dfrac{{{2}^{2}}\times {{2}^{2}}}{{{3}^{2}}}} \\
 & \Rightarrow \sqrt{E}={{\left( \dfrac{{{2}^{2}}\times {{2}^{2}}}{{{3}^{2}}} \right)}^{\dfrac{1}{2}}} \\
\end{align}\]
Using the formula \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\] we get,
\[\begin{align}
  & \Rightarrow \sqrt{E}=\left( \dfrac{{{2}^{2\times \dfrac{1}{2}}}\times {{2}^{2\times \dfrac{1}{2}}}}{{{3}^{2\times \dfrac{1}{2}}}} \right) \\
 & \Rightarrow \sqrt{E}=\left( \dfrac{2\times 2}{3} \right) \\
 & \Rightarrow \sqrt{E}=\dfrac{4}{3} \\
\end{align}\]

Hence, the square root of $\dfrac{48}{27}$ is $\dfrac{4}{3}$.

Note: Note that here we have simplified the given expression and then found the square root. You can also do the reverse process, that is first find the square root and then simplify. In that case you will get $\sqrt{3}$ in the numerator and denominator which will get cancelled and you will get the same answer. Remember all the formulas of exponents to make calculations easier.