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Find the square root of the following number by factorization: 256

seo-qna
Last updated date: 21st Feb 2024
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IVSAT 2024
Answer
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Hint:
Here, we will do the prime factorization of the given number. Factorization is a method of writing an original number as the product of its various factors. We will consider a pair of the same numbers as a single number because we are required to find the square root. Hence, multiplying the remaining factors, we would find the required square root of the given number.

Complete step by step solution:
In this question, first of all, we are required to do the prime factorization of 256.
Prime numbers are those numbers which are greater than 1 and have only two factors, i.e. factor 1 and the prime number itself.
Hence, prime factorization is a method in which we write the original number as the product of various prime numbers.
Therefore, prime factorization of 256 is:
We can see that 256 is an even number, so dividing it by the least prime number 2, we get
\[256 \div 2 = 128\]
Dividing 128 by 2, we get
\[128 \div 2 = 64\]
Dividing 64 by 2, we get
\[64 \div 2 = 32\]
Dividing 32 by 2, we get
\[32 \div 2 = 16\]
Dividing 16 by 2, we get
\[16 \div 2 = 8\]
Dividing 8 by 2, we get
\[8 \div 2 = 4\]
Dividing 4 by 2, we get
\[4 \div 2 = 2\]
Now as we get the quotient as a prime number, so we will not divide the number further.
Hence, 256 can be written as:
\[256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
Now, since we are required to find the square root, we would take only one prime number out of a pair of the same prime numbers.
\[ \Rightarrow \sqrt {256} = 2 \times 2 \times 2 \times 2\]
\[ \Rightarrow \sqrt {256} = 16\]
Hence, the square root of 256 is 16.

Therefore, the required answer is 16.

Note:
If in the question, it was not mentioned that we are required to use the factorization method, then we could have used the ‘Repeated Subtraction Method’ in which we subtract \[1,3,5,...\] from the number in every next step till we get a \[0\] and the number of steps is equal to the square root of the number.