Question

How do you simplify $\sqrt {\dfrac{{25}}{{16}}} $ ?

Answer 1

Hint: Here, we have to calculate the square roots of fraction. So, firstly find the square root of the numerator and then the square root of the denominator. Dividing the square root of the numerator by the square root of the denominator, we get the required result.

Complete step-by-step solution:
Given, we have to simplify $\sqrt {\dfrac{{25}}{{16}}} $.
We can also write $\sqrt {\dfrac{{25}}{{16}}} $ as $\dfrac{{\sqrt {25} }}{{\sqrt {16} }}$.
Now, we have to separately find the square roots of the numerator and the denominator.
The square root of the numerator$\sqrt {25} $ by prime factorization method is given by
The prime factor of $25$ is $5 \times 5$.
So, $\sqrt {25} = \sqrt {5 \times 5} = 5$.
Thus, the square root of the numerator $\sqrt {25} $ is $5$.
The square root of the denominator$\sqrt {16} $ by prime factorization method is given by
The prime factor of $16$ is $2 \times 2 \times 2 \times 2$.
So, $\sqrt {16} = \sqrt {2 \times 2 \times 2 \times 2} = 2 \times 2 = 4$.
Thus, the square root of the denominator $\sqrt {16} $ is $4$.
 Now, dividing the square root of the numerator by the square root of the denominator we get
\[\sqrt {\dfrac{{25}}{{16}}} = \dfrac{{\sqrt {25} }}{{\sqrt {16} }} = \dfrac{5}{4} = 1.25\]

Thus, the required value of $\sqrt {\dfrac{{25}}{{16}}} $ is $1.25$.

Note: Method of finding the square root by Prime factorization:
In this method firstly, we have to find the prime factors of the given number and arrange the prime factors in increasing order then make a pair of the same number and multiplying a number from each pair gives the square root of the given number.
Similarly, we can calculate the cube root of a given fraction with only difference is that in spite of finding the square root of the numerator and the denominator we have to calculate the cube roots of the numerator and the denominator.