Question

How do you find the square root of \[\dfrac{25}{9}\]?

Answer 1

Hint: Using the property of the square root that is the square root of the fraction is equal to the square root of the numerator divided by the square root of the denominator.
\[\Rightarrow \sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}\].

Complete step by step solution:
As we know that the according to the property of the square root of a fraction, can be calculated by distributing the square root to numerator and denominator both.
\[\Rightarrow \sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}\]
Now comparing the given question with this identity
\[a=25\] and \[b=9\]
\[\Rightarrow \sqrt{\dfrac{25}{9}}=\dfrac{\sqrt{25}}{\sqrt{9}}\text{ ---(1)}\]
Now, we have to calculate the square root of \[25\] and \[9\]
Since the square root means the base having the exponential power \[\dfrac{1}{2}\]
\[\Rightarrow \sqrt{25}={{(25)}^{\dfrac{1}{2}}}\]
Also \[25\] can be written as \[{{5}^{2}}\] in exponential form
\[\Rightarrow \sqrt{25}={{(25)}^{\dfrac{1}{2}}}={{({{5}^{2}})}^{\dfrac{1}{2}}}\]
We also know that, \[{{({{a}^{m}})}^{n}}={{a}^{mn}}\]
\[\Rightarrow \sqrt{25}={{(25)}^{\dfrac{1}{2}}}={{({{5}^{2}})}^{\dfrac{1}{2}}}=5\]
Similarly,
\[\Rightarrow \sqrt{9}={{(9)}^{\dfrac{1}{2}}}={{({{3}^{2}})}^{\dfrac{1}{2}}}=3\]
Now from equation \[(1)\]
\[\Rightarrow \sqrt{\dfrac{25}{9}}=\dfrac{\sqrt{25}}{\sqrt{9}}=\dfrac{5}{3}\]

Hence the square root of \[\dfrac{25}{9}\] is \[\dfrac{5}{3}\].

Note: To calculate the square root of a fraction, the square root will be distributed to both numerator and denominator and then simplify the final value obtained by these distributed square roots.