Question

What is the square root of $\dfrac{125}{5}$?

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 answer:
Here, we have been provided with the expression $\dfrac{125}{5}$ and we are asked to find its square root. Let us assume the given expression as E. So, we have,
\[\Rightarrow E=\dfrac{125}{5}\]
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. Since the number in the denominator is already a prime so we will leave it and find the prime factors of the numerator only.
\[\Rightarrow 125=5\times 5\times 5\]
Substituting this in expression E we get,
\[\Rightarrow E=\dfrac{5\times 5\times 5}{5}\]
Cancelling the common factors we get,
\[\Rightarrow E=5\times 5\]
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,
\[\Rightarrow E={{5}^{2}}\]
Now, taking square root both the sides we get,
\[\begin{align}
  & \Rightarrow \sqrt{E}=\sqrt{{{5}^{2}}} \\
 & \Rightarrow \sqrt{E}={{\left( {{5}^{2}} \right)}^{\dfrac{1}{2}}} \\
\end{align}\]
Using the formula \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\] we get,
\[\begin{align}
  & \Rightarrow \sqrt{E}={{5}^{2\times \dfrac{1}{2}}} \\
 & \Rightarrow \sqrt{E}=5 \\
\end{align}\]

Hence, the square root of $\dfrac{125}{5}$ is 5.

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. You must know how to find prime factors of a number because in many cases the given number will be very large and in such a case we would have no option other than finding the prime factors. Remember all the formulas of exponents to make calculations easier.