Question

Find the cube root of 74088.

Answer 1

Hint: We will find the cube root by the prime factorization method. Now, we will understand the definition of prime factorization which is given as prime numbers when multiplied together to make the original number. Then we will find factors of the number and we will group it into triplets as we have to find the cube root of the number. For example, if we want to find the cube root of 27 then we can write it as $3 \times 3 \times 3$. So, the cube root is 3. Similarly, we will get the cube root of the number.

Complete step-by-step solution:
Here, we will first understand the prime factorization method.
Prime Factorization is a method of finding which. prime numbers when multiplied together make the original number. For example: let us take number 12. So, the prime number which when multiplied give number 12 is $2 \times 2 \times 3$.
Similarly, we will first find prime factors of number 4096. So, we will divide this number using a prime number only. We will get as,
\[\begin{array}{l}
2\left| \!{\underline {\,
  {74088} \,}} \right. \\
2\left| \!{\underline {\,
  {37044} \,}} \right. \\
2\left| \!{\underline {\,
  {18522} \,}} \right. \\
3\left| \!{\underline {\,
  {9261} \,}} \right. \\
3\left| \!{\underline {\,
  {3087} \,}} \right. \\
3\left| \!{\underline {\,
  {1029} \,}} \right. \\
7\left| \!{\underline {\,
  {343} \,}} \right. \\
7\left| \!{\underline {\,
  {49} \,}} \right. \\
7\left| \!{\underline {\,
  7 \,}} \right.
\end{array}\]
So, the prime factorization of 74088 is,
$ \Rightarrow 74088 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 7 \times 7 \times 7$
Now, we have to find the cube root of the number. So, we will pair the same digit in a group of three. So, we get as
$ \Rightarrow 4096 = \underline {2 \times 2 \times 2} \times \underline {3 \times 3 \times 3} \times \underline {7 \times 7 \times 7} $
Thus, we will multiply a single digit from all the four triplets,
$ \Rightarrow 2 \times 3 \times 7 = 42$

Hence, the cube root of 74088 is 42.

Note: It can be solved by another method by estimation.
We can see that we have two groups i.e. $74$ and $088$.
If any number is ending in 8, then its cube root's digit must be ending in 2.
Now, 74 lies between,
$ \Rightarrow {4^3} \le 74 \le {5^3}$
The minimum number is 4.
Hence, the cube root of 74088 is 42.