Question

The square root of \[\dfrac{{441}}{{961}}\] is:
A) \[\dfrac{{21}}{{39}}\]
B) \[\dfrac{{37}}{{21}}\]
C) \[\dfrac{{21}}{{31}}\]
D) \[\dfrac{{11}}{{13}}\]

Answer 1

Hint:
Here, we will first use the property of square roots of rational numbers, \[\sqrt {\dfrac{a}{b}} = \dfrac{{\sqrt a }}{{\sqrt b }}\] in the given number and then compute the square roots of the numerator and denominator separately using prime factorization to find the required value.

Complete step by step solution:
We are given that the number is \[\dfrac{{441}}{{961}}\].
Taking the square root in the given number, we get
\[ \Rightarrow \sqrt {\dfrac{{441}}{{961}}} \]
Using the property of square roots of rational numbers, \[\sqrt {\dfrac{a}{b}} = \dfrac{{\sqrt a }}{{\sqrt b }}\] in the above equation, we get
\[ \Rightarrow \dfrac{{\sqrt {441} }}{{\sqrt {961} }}\]
Now, let us compute the square roots of the numerator and denominator separately.
First, take the prime factorization of the denominator \[\sqrt {441} \], we get
\[
   \Rightarrow \sqrt {441} = \sqrt {3 \times 3 \times 7 \times 7} \\
   \Rightarrow \sqrt {441} = 3 \times 7 \\
   \Rightarrow \sqrt {441} = 21{\text{ ......eq.(1)}} \\
 \]
Now we will take the prime factorization of the denominator \[\sqrt {961} \], we get
\[
   \Rightarrow \sqrt {961} = \sqrt {31 \times 31} \\
   \Rightarrow \sqrt {961} = 31{\text{ ......eq.(2)}} \\
 \]
Taking (1) in numerator and (2) in denominator to find the required value, we get
\[ \Rightarrow \sqrt {\dfrac{{441}}{{961}}} = \dfrac{{21}}{{31}}\]

Hence, option C is correct.

Note:
In solving these types of questions, you should be familiar with the steps to find the square root using the prime factorization method. A rational numbers are those numbers which can be written in the form of \[\dfrac{p}{q}\], where \[p\] is numerator, \[q\] is denominator, \[q \ne 0\] and both are integers.
 We can also verify our solution by finding the square of the obtained square root.
\[\dfrac{{21}}{{31}} \times \dfrac{{21}}{{31}} = \dfrac{{441}}{{961}}\]
Hence, \[\dfrac{{21}}{{31}}\] is the square root of the number \[\dfrac{{441}}{{961}}\].