Question

Find the square root of the following number by the prime factorization method.
1936.

Answer 1

Hint: To solve this question, we will use the concept of prime factorization. The fundamental theorem of arithmetic states that every composite number can be expressed or we can say that it is factorized as a product of prime numbers, and this factorization is unique except for the order in which the prime factors occur.

Complete step-by-step answer:
The numbers that are divisible by only two numbers that are 1 and by themselves are called prime numbers.
Given that, 1936.
We have to find its square root by a prime factorization method.
First, we have to make the prime factors of 1936.
\[
   \Rightarrow 1936 = 2 \times 968 \\
   \Rightarrow 1936 = 2 \times 2 \times 484 \\
   \Rightarrow 1936 = 2 \times 2 \times 2 \times 242 \\
   \Rightarrow 1936 = 2 \times 2 \times 2 \times 2 \times 121 \\ \]
\[ \Rightarrow 1936 = 2 \times 2 \times 2 \times 2 \times 11 \times 11\] ………… (i)
These are the prime factors of 1936.
Now, we will find out its square root.
Taking square root on both sides of equation (i), we will get
\[
   \Rightarrow \sqrt {1936} = \sqrt {2 \times 2 \times 2 \times 2 \times 11 \times 11} \\
   \Rightarrow \sqrt {1936} = \sqrt {\left( {2 \times 2} \right) \times \left( {2 \times 2} \right) \times \left( {11 \times 11} \right)} \\
   \Rightarrow \sqrt {1936} = \sqrt {{{\left( {2 \times 2} \right)}^2} \times {{\left( {11} \right)}^2}} \\
   \Rightarrow \sqrt {1936} = 2 \times 2 \times 11 \\
   \Rightarrow \sqrt {1936} = 44 \\ \]
Here, we can see that the square root of 1936 by the prime factorization method is 44.

Note: Whenever we ask this type of question, first, we have to remember what are prime numbers? And also, we should know the prime factorization method. Then we have to find out the prime factors of the given number and after that we will do the square root of that number and its prime factors. By solving this, we will get the required answer.