Question

Find the square-root of $\dfrac{{441}}{{961}}$
A. $\dfrac{{21}}{{39}}$
B. $\dfrac{{37}}{{21}}$
C. $\dfrac{{21}}{{31}}$
D. $\dfrac{{11}}{{13}}$

Answer 1

Hint: Square-root of a number (A) is the number which is multiplied by itself to produce the number (A). To find the square-root of number one can use the prime factorization method. Find the prime factors of the number and make pairs of the same factors. Write one factor from each pair and multiply them. This product is the required square-root.

Complete step by step solution: We have to find the square-root of the fraction $\dfrac{{441}}{{961}}$.
First, we have to reduce the fraction to its lowest form
To do this we divide the numbers of the numerator and denominator into their prime factors.
We know that $441 = 3 \times 3 \times 7 \times 7$ and $961 = 31 \times 31$ so we can write-
$ \Rightarrow \sqrt {\dfrac{{441}}{{961}}} = \sqrt {\dfrac{{3 \times 3 \times 7 \times 7}}{{31 \times 31}}} $
Now make the pair of similar factors so that both the factors in the pair are the same and equal number.
Here we see that $\left( {3 \times 3} \right)$ and $\left( {7 \times 7} \right)$ are two pairs in numerator such that both factors are equal in the pairs.
And in denominator $\left( {31 \times 31} \right)$ is also a pair of such factors that are equal.
So we can write-
$ \Rightarrow \sqrt {\dfrac{{441}}{{961}}} = \sqrt {\dfrac{{\left( {3 \times 3} \right) \times \left( {7 \times 7} \right)}}{{\left( {31 \times 31} \right)}}} $
Now take on factor from each pair out of the square-root
$ \Rightarrow \sqrt {\dfrac{{441}}{{961}}} = \dfrac{{3 \times 7}}{{31}}$
Now multiply the factors to find the product of the factors obtained from each pair. This product is the square-root of the given fraction.
$ \Rightarrow \sqrt {\dfrac{{441}}{{961}}} = \dfrac{{21}}{{31}}$

Hence the correct answer is C.

Note: Prime factors are the numbers which are prime numbers. A prime number is a number which can be divided by itself and $1$ only. Prime factorization method is also used to find HCF( highest common factor) and LCM (lowest common factor) of a number.