Question

# By splitting into prime factors, find the square root of 5184.

Hint: The given problem is related to square and square root roots. To determine the square root of a number, express the number in terms of the product of its prime factors. This is the method of prime factorization.

Before proceeding with the solution, let’s understand the concept of prime factorization. A prime number is a number which is not divisible by any other number except 1 and itself. Any number can be expressed as a product of prime numbers. All the prime numbers, which when multiplied, give a product equal to a number (say x) are called the prime factors of the number x. To write the prime factors of a number, we should always start with the smallest prime number, i.e. 2 and check divisibility. If the number is divisible by the prime number, then we write the number as a product of the prime number and another number, which will be the quotient when the given number is divided by the prime number. Then, we take the quotient and repeat the same process. This process is repeated till we are left with 1 as the quotient.
Now, coming to the question, we are asked to find the square root of 5184. We will use the method of prime factorization. So, first we will express 5184 as the product of prime numbers.
Now, 5184 is an even number. So, we can write 5184 as $5184=2\times 2592$ . Now, 2592 is also an even number. So, we can write 2592 as $2592=2\times 1296$ . Similarly, we can write 1296 as $1296=2\times 648$ . Now, 648 is also an even number number. So, it is divisible by 2. Therefore, 648 can be written as $648=2\times 324$ . Again, 234 being an even number can be written as $324=2\times 162$ . Further, 162 can be written as $162=2\times 81$ . Now, 81 is an odd number. The sum of digits of 81 = 1 + 8 = 9, which is divisible by 3. So, 81 is divisible by 3. So, we can write 81 as $81=3\times 27$ . Again, the sum of digits of 27 is 2 + 7 = 9, which is divisible by 3. So, we can write 27 as $27=3\times 9$ . We know, 9 is the square of 3. So, we can write 9 as $9=3\times 3$ . So, $5184=2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3\times 3$ . So, the square root of 5184 can be written as $\sqrt{5184}=2\times 2\times 2\times 3\times 3=72$ . Hence, the square root of 5184 is 72.

Note: While calculating square roots and cube roots, prime factorization is the easiest method. But it takes time. Hence, other methods should also be learnt, so that they can be used while solving problems in cases where time plays an important role.