Question

Find the square root of a number 9604 by Prime Factorization Method.

Answer 1

Hint: In this question first of all write prime factors of 9604 by using the prime factorization method. Then take square roots on both sides to get the square root of 9604 which is the required solution.

Complete step-by-step answer:
To find prime factors of 9604 follow the below steps
Step 1. As the first prime i.e., 2 is a factor of 9604, we can write 9604 as
\[ \Rightarrow 9604 = 2 \times 4802\]
Step 2. As the first prime i.e., 2 is a factor of 4802, we can write 9604 as
\[ \Rightarrow 9604 = 2 \times 2 \times 2401\]
Step 3. As the 7 is the next prime factor of 2401, we can write 9604 as
\[ \Rightarrow 9604 = 2 \times 2 \times 7 \times 343\]
Step 4. As the 7 is the next prime factor of 343, we can write 9604 as
\[ \Rightarrow 9604 = 2 \times 2 \times 7 \times 7 \times 49\]
Step 5. As the 7 is the next prime factor of 49, we can write 9604 as
\[ \Rightarrow 9604 = 2 \times 2 \times 7 \times 7 \times 7 \times 7\]
So, by using method of prime factorization we have
\[9604 = {2^2} \times {7^2} \times {7^2}\]
Taking square roots on both sides, we get
\[
   \Rightarrow \sqrt {9604} = \sqrt {{2^2} \times {7^2} \times {7^2}} \\
   \Rightarrow \sqrt {9604} = \sqrt {{{\left( {2 \times 7 \times 7} \right)}^2}} \\
   \Rightarrow \sqrt {9604} = 2 \times 7 \times 7 \\
  \therefore \sqrt {9604} = 98 \\
\]
Thus, the square root of 9604 is 98.

Note: As 9604 is a positive integer the square of 9604 is also a positive integer. We can verify our answer as if we get 9604 by squaring 98, then our answer is correct. Only consider the prime factors of 9604 as we are doing prime factorization for 9604.