Find the cube root of 5382.

Hint: In this question, we need to factorise the given number as the cube of some number. Then after expressing it as the cube of some number we need to apply the cube root.

Complete step-by-step answer:

Cube of a number is the number multiplied by itself thrice.
Cube of n is given by:
\[n\times n\times n={{n}^{3}}\]
Cube root of a number is another number that, when you multiply it by itself three gives you the original number.
Cube root of n is given by:
\[\sqrt[3]{n}\] or \[{{n}^{\dfrac{1}{3}}}\]
Now, let us first factorise the given number in the question.
\[\Rightarrow 5832=2\times 2\times 2\times 9\times 9\times 9\]
Let us now represent them as cube terms.
As we already know that \[n\times n\times n={{n}^{3}}\]
\[\Rightarrow 5832={{2}^{3}}\times {{9}^{3}}\]
Now this can be rewritten as:
\[\Rightarrow 5832={{\left( 2\times 9 \right)}^{3}}\]
On further simplification we get,
\[\Rightarrow 5832={{18}^{3}}\]
Now, by applying the cube root on both the sides we get,
\[\Rightarrow \sqrt[3]{5832}=\sqrt[3]{{{18}^{3}}}\]
Now, the right hand side of the above equation can be rewritten as:
\[\Rightarrow \sqrt[3]{5832}={{\left( {{18}^{3}} \right)}^{\dfrac{1}{3}}}\]
Now, from this on further simplification we get,
\[\therefore \sqrt[3]{5832}=18\]
Hence, the cube root of 5832 is 18.

Note: It is important to note that the cube root of a number is unique. So, a particular number can have only one real cube root which can be found either by factoring or by trial and error. But it is difficult to predict the possible number in one go by using trial and error so it is preferable to use the factorisation method.
While solving the question we need to be careful while factoring because missing any of the terms results in less number of factors. So, it cannot be expressed as a cube of a number and so we cannot find the cube root for that number.