Question

Find the cube root of \[27000\] by prime factorization method.

Answer 1

Hint:
In this question first we will find the prime factor of the number given to us and then we will try to make the triplets of the prime factors. Now, if we get the triplets of the prime numbers then it will be a perfect cube. Now, if we multiply this perfect cube we will get the perfect cube root of the given number.

Complete step by step solution:
We know the process of squaring. Therefore, the process of cubing is similar to it. The only difference is that the number is multiplied three times instead of two times. For example ${7^3} = 7 \times 7 \times 7 = 343$ .
Now, we know that in the prime factorization method we have to find the prime factors of the given number.
We can write \[27000\] as \[2 \times 13500\] . Now, we will further factorize \[2 \times 13500\] and we can write it as $2 \times 2 \times 2 \times 3375$ . Now find the prime factors of $3375$ . Therefore, $2 \times 2 \times 2 \times 3375$ can be written as $2 \times 2 \times 2 \times 3 \times 1125$ and on further factorization in terms of prime numbers it can be written as $2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 125$ . Finally, we will find the prime factors of $125$ . We know that $125 = 5 \times 5 \times 5$ . Hence, $2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 125$ can be written as $2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5 \times 5 \times 5$ .
Therefore, from the above calculation we can write \[27000\] as the multiplication of its prime factors i.e. $27000 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5 \times 5 \times 5 = {2^3} \times {3^3} \times {5^3}$ .
Now, we got the triplets of$2,\,3$ and $5$ . Therefore, on multiplying each of these prime numbers three times we will get \[27000\] . Therefore, we can write $\sqrt[3]{{27000}} = \sqrt[3]{{{2^3} \times {3^3} \times {5^3}}} = 2 \times 3 \times 5 = 30$ .

Hence, we can say $30$ is the cube root of $27000$.

Note:
We should be able to find the prime factors of the given number and we should be able to make triplets of that prime factors. We should know that if a number is formed by multiplying its factor three times then that factor is the cube root of the given number.