Question

Is 9261 a perfect cube? In case of a perfect cube, find the number whose cube it is.

Answer 1

Hint:
In this question we have to find whether 9261 is a perfect cube or not, hence, we would do the prime factorization of the given number. Then, we would consider a group of triplets of the same numbers as a single number because we are required to find the cube root. If we are able to group each and every factor in any group of triplets then the given number is a perfect cube. But if even a single factor is left out then we can’t find a perfect cube root of the given number. Hence, multiplying the remaining factors, we would be able to find the required cube root of the given number.

Complete step by step solution:
To answer this question, first of all, we should know the below facts:
To find whether a number is a perfect cube or not, we are required to use the prime factorization method.
Now, factorization is a method of writing an original number as the product of its various factors.
Also, prime numbers are those numbers which are greater than 1 and have only two factors, i.e. factor 1 and the prime number itself.
Hence, prime factorization is a method in which we write the original number as the product of various prime numbers.
Now, after doing the prime factorization, we are required to club the triplets of same prime factors together, as a single prime factor and if no factor is left out then the number is a perfect cube.
Therefore, prime factorization of 9261 is:
$\begin{matrix}
   3 & 9261 \\
   3 & 3087 \\
   3 & 1029 \\
   7 & 343 \\
   7 & 49 \\
   7 & 7 \\
   {} & 1 \\
\end{matrix}$
Hence, 9261 can be written as:
\[9261 = 3 \times 3 \times 3 \times 7 \times 7 \times 7\]
Hence, we have expressed the given number as a product of its prime factors.
Now, we would group the triplets of same prime factors together, as a single prime factor:
\[ \Rightarrow 9261 = \left( {3 \times 3 \times 3} \right) \times \left( {7 \times 7 \times 7} \right)\]
Clearly, we are able to group the factors without leaving behind any factor which is not in any group, hence,
 9261 a perfect cube.
Now, solving further, by converting the group of triplets as a single prime factor, we get,
\[\sqrt[3]{{9261}} = 3 \times 7 = 21\]
Hence,
 9261 is the cube of \[21\].

Therefore, the required answer is:
Yes,
9261 is a perfect cube and the cube root of 9261 is 21.

Note:
Prime numbers are those numbers which are greater than 1 and have only two factors, i.e. factor 1 and the prime number itself. Hence, prime factorization is a method in which we write the original number as the product of various prime numbers. Also, the most common mistake which we could do is grouping the factors in a pair rather than a triplet and finding the square root instead of cube root. Hence, it should be kept in mind while solving what is asked to find in the question.