Question

How do you find $ \dfrac{\sqrt{25}}{\sqrt{81}} $ ?

Answer 1

Hint: Students find such questions very easy and this the reason they create silly mistakes in such questions. In this question first, we rationalize the given fraction by multiplying and dividing by $ \sqrt{81} $ and then solving the question by using the perfect square concept as 25 and 81 both are perfect squares of 5 and 9 respectively.

Complete step by step answer:
According to the question, we have been asked to simplify the value of $ \dfrac{\sqrt{25}}{\sqrt{81}} $
Now, to solve this question, we will multiply the numerator and the denominator by $ \sqrt{81} $
Therefore, we get
 $ \begin{align}
  & \dfrac{\sqrt{25}}{\sqrt{81}}\times \dfrac{\sqrt{81}}{\sqrt{81}} \\
 & \Rightarrow \dfrac{\sqrt{25\times 81}}{\sqrt{81\times 81}} \\
 & \\
\end{align} $
We know that 25 can be expressed as 5 × 5 and 81 can be expressed as 9 × 9. So, we get
 $ \begin{align}
  & \Rightarrow \dfrac{\sqrt{5\times 5\times 9\times 9}}{81} \\
 & \Rightarrow \dfrac{\sqrt{5\times 5}\times \sqrt{9\times 9}}{81} \\
 & \\
\end{align} $
And we know that the square root of the product of two same numbers gives the same number in return. Therefore, we get
 $ \begin{align}
  & \Rightarrow \dfrac{5\times 9}{81} \\
 & \Rightarrow \dfrac{5\times 9}{9\times 9} \\
\end{align} $
And we know that the same terms in numerator and denominator of a fraction cancels out. Therefore, after cancelling 9 from both numerator and denominator we get
 $ \Rightarrow \dfrac{5}{9} $
Therefore, we get a simplified expression as
 $ \dfrac{\sqrt{25}}{\sqrt{81}}=\dfrac{5}{9} $ .

Note:
The most important thing here to note is that students can solve this directly but that should not be done. We should clearly mention all the steps which will be helpful for the examiner to understand the answer. The possible mistake one can make while solving the question in hurry is by writing $ \sqrt{81} $ = 3 instead of 9 which will give us the wrong answer and henceforth we will lose our marks.