Question

Find the cube root of the following number by prime factorization number method: 110592.
(a) 48
(b) 38
(c) 58
(d) 68

Answer 1

Hint: First, we will understand the definition of prime factorization which is given as prime numbers when multiplied together make the original number. Then we will find factors of the number 110592 and we will group it into pairs of 3 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 $ \underline{3\times 3\times 3} $ and so the answer is number 3. Similarly, by doing this we will find the answer.

Complete step-by-step answer:
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 110592. So, we will divide this number using a prime number only. We will get as
 $ \begin{align}
  & 2\left| \!{\underline {\,
  110592 \,}} \right. \\
 & 2\left| \!{\underline {\,
  55296 \,}} \right. \\
 & 2\left| \!{\underline {\,
  27648 \,}} \right. \\
 & 2\left| \!{\underline {\,
  13824 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6912 \,}} \right. \\
 & 2\left| \!{\underline {\,
  3456 \,}} \right. \\
 & 2\left| \!{\underline {\,
  1728 \,}} \right. \\
 & 2\left| \!{\underline {\,
  864 \,}} \right. \\
 & 2\left| \!{\underline {\,
  432 \,}} \right. \\
 & 2\left| \!{\underline {\,
  216 \,}} \right. \\
 & 2\left| \!{\underline {\,
  108 \,}} \right. \\
 & 2\left| \!{\underline {\,
  54 \,}} \right. \\
 & 3\left| \!{\underline {\,
  27 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
\end{align} $
We get as $ 110592=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3 $
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 $ 110592=\underline{2\times 2\times 2}\times \underline{2\times 2\times 2}\times \underline{2\times 2\times 2}\times \underline{2\times 2\times 2}\times \underline{3\times 3\times 3} $
Thus, we will multiply a single digit from all the five pairs i.e. $ 2\times 2\times 2\times 2\times 3=48 $ .
Thus, the cube root of 110592 is 48.
So, the correct answer is “Option A”.

Note: Another method of solving is by option method. If we take option (b) i.e. 38 and we multiply it 3 times, then we should get an answer equal to 110592. So, by this also we can directly find out which number on the cube gives the number in question. Be careful in the prime factorization method. Only take prime numbers and then solve so not take any other numbers like 4,6,8… so on which are not prime and get the answer. It will be the wrong method.