Question

Find the cube root of 27000.

Answer 1

Hint: We will factorize the given number with prime numbers. Then we will group these factors into sets of three, since we have to find the cube root. Then we will get the cube root of 27000.

Complete step by step answer:
We have to find the cube root of 27000. We will first factor it by 2. So, we have the following,
\[27000=2\times 13500\]
We can again factor 13500 by 2 in the following manner,
\[27000=2\times 2\times 6750\]
Again, factoring 6750 by 2, we get
\[27000=2\times 2\times 2\times 3375\]
Now, we will factor 3375 by 5. We will get the following expression,
\[27000=2\times 2\times 2\times 5\times 675\]
Again, factoring 675 by 5, we have
\[27000=2\times 2\times 2\times 5\times 5\times 135\]
Factoring 135 by 5, we get the following expression,
\[27000=2\times 2\times 2\times 5\times 5\times 5\times 27\]
Now, we have to factor 27. We will factor 27 by 3 in the following manner,
\[27000=2\times 2\times 2\times 5\times 5\times 5\times 3\times 9\]
Factoring 9 by 3, we get
\[27000=2\times 2\times 2\times 5\times 5\times 5\times 3\times 3\times 3\]
Now, we see the factors in groups of three in the above expression. Hence,
\[27000={{2}^{3}}\times {{5}^{3}}\times {{3}^{3}}\]
The above expression can be rewritten as follows,
\[\begin{align}
  & 27000={{(2\times 5\times 3)}^{3}} \\
 & ={{30}^{3}}
\end{align}\]

Hence, the cube root of 27000 is 30.

Note: In this question, the factorization should be done carefully. The grouping of factors is the key step to achieve the desired answer. There is another way to approach this question. We can split 27000 as $27000=27\times 1000$. And we know that 27 is the cube of 3 and 1000 is the cube of 10. So, 27000 is the cube of $3\times 10$, that is 30.