Question

Find the square root of 256 by prime factorization method.

Answer 1

Hint: We are asked to find the square root of a number using prime factorization. For that we first need to repeatedly divide the number by the smallest prime possible and we need to that till it becomes 1. So, in short we need to find the prime numbers which when multiplied will give the number whose prime factorization is needed to be found out. After finding that we will try to convert that as a square of the primes and then we apply the root function on that.

Complete step by step answer:
We first need to find the prime factorization of a number, for which we will continuously divide the number by the smallest prime possible. For example, if the number is 6 then we first find the smallest prime that divides 6, which is 2 and then we divide it further by 3 because that is the smallest prime which divides 6 after 2. And now only 1 is left which means the prime factorization of 6 is as follows:
$6=2\times 3\times 1$
Now, we have 256. The smallest prime number that divides 256 is 2, so we have:
$256=2\times 128$
Now, we further factorize 128. The smallest prime number that divides 128 is 2 again, so till now we have reached:
$256=2\times 2\times 64$
Again, 2 divide 64, so we have:
$256=2\times 2\times 2\times 32$
$\Rightarrow 256=2\times 2\times 2\times 2\times 16$
$\Rightarrow 256=2\times 2\times 2\times 2\times 2\times 8$
$\Rightarrow 256=2\times 2\times 2\times 2\times 2\times 2\times 4$
$\Rightarrow 256=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$
So the prime factorization of 256 has been obtained.
Now, we collect the terms so that we are able to form a square.
\[256={{\left( 2\times 2\times 2\times 2 \right)}^{2}}\]
$\Rightarrow 256={{\left( 16 \right)}^{2}}$
Since, 256 is the square of 16, the square root of 256 is 16. Hence, the square root has been found out.

Note: You need to be very careful while giving the factors, because only prime factors are allowed while finding the prime factorization. For example, if you write $256=2\times 2\times 2\times 32$ then that would be wrong because 32 is not a prime number. So, you need to be aware while listing out the prime factors.