Question

Find the square root of 6724 by prime factorization method.

Answer 1

Hint:
Here we will first find the prime factors of the given number using the prime factorization method. We will then write the given number in terms of prime factors. Then we will use different properties of exponential function to get the required square root of the given number.

Complete step by step solution:
The given number is 6724.
We will first find the prime factors of the given number using the prime factorization method.
We will first divide the 6724 by the least prime number 2.
\[{\text{6724}} \div 2 = 3362\]
Again we will divide the obtained number by the least prime number 2.
\[3362 \div 2 = 1681\]
Again we will divide the obtained number by the next least prime number 41. Therefore, we get
\[1681 \div 41 = 41\]
As we can see that 41 is a prime number. So we can’t factorize that number further.
Therefore, the prime factors of the given number, 6724 are 2, 2 , 41 and 41.
We can write 6724 as:
$6724 = 2 \times 2 \times 41 \times 41$…………………$\left( 1 \right)$
Now, we will find the square of the given number 6724.
Taking square root on both sides of equation $\left( 1 \right)$, we get
$ \Rightarrow \sqrt {6724} = \sqrt {2 \times 2 \times 41 \times 41} $
Rewriting the equation, we get
 $ \Rightarrow \sqrt {6724} = \sqrt {{2^2} \times {{41}^2}} $
Now, we will write the square root in the power form. Therefore, we get
$ \Rightarrow \sqrt {6724} = {\left( {{2^2} \times {{41}^2}} \right)^{\dfrac{1}{2}}}$
Using the property of exponent ${{\left( {{x}^{m}} \right)}^{n}}={{x}^{m\times n}}$, we get
$ \Rightarrow \sqrt {6724} = {2^{2 \times \dfrac{1}{2}}} \times {41^{2 \times \dfrac{1}{2}}}$
On further simplifying the terms, we get
$ \Rightarrow \sqrt {6724} = 2 \times 41$
On multiplying the terms, we get
$ \Rightarrow \sqrt {6724} = 82$

Therefore, the required square root of the given number i.e. 6724 is equal to 82..

Note:
Prime factorization is a method by which we factorize numbers in terms of prime factors. Prime factors of a number are defined as the set of prime numbers which when multiplied together will give the actual number. We know that 2 is the only even prime number and also the least prime number among all. Therefore, in order to find the prime factors, we will first divide the number by 2. If the number is not divisible by 2, then we will divide by next least prime number 3 and continue the process until we reach a situation where the number obtained after dividing is prime number only.