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

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 $2\times 2\times 3$.
\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. $2\times 2\times 2\times 2\times 3=48$ .