Question

Find the cube root of 74088.

Answer 1

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

Complete step-by-step answer:

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.

For example: Consider the number 51. It is an even number. So, it is not divisible by 2. The sum of the digits of 51 is 5 + 1 = 6. Hence, 51 is divisible by 3. Now, $51=3\times 17$ . Now, we take 17. We know, 17 is a prime number. Hence, the prime factors of 51 are 3 and 17.

Now, coming to the question, we are asked to find the cube root of 74088. We will use the method of prime factorization. So, first we will express 74088 as the product of prime numbers.

Now, 74088 is an even number. So, we can write 74088 as $74088=2\times 37044$ . Now, 37044 is also an even number. So, we can write 37044 as $37044=2\times 18522$ . Similarly, we can write 18522 as $18522=2\times 9261$ .

 Now, 9261 is an odd number. So, it is not divisible by 2. The sum of digits of 9261 = 9 + 2 + 6 + 1 = 18, which is divisible by 3. So, 9261 is divisible by 3. So, we can write 9261 as $9261=3\times 3087$ . Again, the sum of digits of 3087 is 3 + 0 + 8 + 7 = 18, which is divisible by 3. So, we can write 3087 as $3087=3\times 1029$ .

 Again, the sum of digits of 1029 is 1 + 0 + 2 + 9 = 12, which is divisible by 3. So, we can write $1029=3\times 343$ . We know, 343 is the cube of 7. So, we can write 343 as $343=7\times 7\times 7$ . So, $74088=2\times 2\times 2\times 3\times 3\times 3\times 7\times 7\times 7$ . So, the cube root of 74088 can be written as $\sqrt[3]{74088}=2\times 3\times 7=42$ .

Hence, the cube root of 74088 is 42.




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.