Question

Find the square root of the number 7744.

Answer 1

Hint: First reduce the given number as the multiple of powers of its prime factors. This will be in the form of $N = {p^a} \times {q^b} \times {r^c} \times ....$, where \[p,{\text{ }}q,{\text{ }}r,...\] are prime factors of $N$. Then for finding the square root of $N$, divide the powers of prime numbers by 2. The number left after dividing the powers by 2 will be the answer.

Complete step-by-step solution:
According to the question, the given number is 7744. Its square root is to be determined. We will use a prime factorization method to find it.
On applying the prime factorization method, first we will reduce the number as the multiple of powers of its prime factors.
As we can clearly see that the number 7744 is divisible by 11, which is a prime number. So we can write it as:
$ \Rightarrow 7744 = 11 \times 704$
The number left i.e. 704 can also be divided by 11. Dividing it by 11, we’ll get:
\[ \Rightarrow 7744 = 11 \times 11 \times 64\]
We know that 64 can be written as 6 raise to the power 6 i.e. ${2^6}$. Doing this and writing 11’s also in powers, we’ll get:
\[ \Rightarrow 7744 = {11^2} \times {2^6}\]
Now, for finding the square root of 7744, we’ll divide the powers by 2 on both sides. So we have:
\[
   \Rightarrow {\left( {7744} \right)^{\dfrac{1}{2}}} = {11^{\dfrac{2}{2}}} \times {2^{\dfrac{6}{2}}} \\
   \Rightarrow \sqrt {7744} = 11 \times {2^3} = 11 \times 8 \\
   \Rightarrow \sqrt {7744} = 88
 \]

Thus the square root of the number 7744 is 88.

Note: In case we have to find the square root of any number by the same method, we follow the same steps. First we reduce the given number as the multiple of powers of its prime factors. And in the last step, we divide the powers by 2 instead of 3. Similar is the case with other roots. If the fourth root is required, divide the powers by 4 in the last step and so on.