Find the cube of the following binomial expressions:
$\left( {4 - \dfrac{1}{{3x}}} \right)$

Hint – In this question we simply need to find the cube of the given binomial expression so simply use the direct formula for ${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}$ to get the answer.

Complete step-by-step answer:
Given binomial expression is
$\left( {4 - \dfrac{1}{{3x}}} \right)$
Now we have to find out the cube of this expression.
$ \Rightarrow {\left( {4 - \dfrac{1}{{3x}}} \right)^3}$
Now as we know ${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}$ so, use this property in above equation and expand we have,
 $ \Rightarrow {\left( {4 - \dfrac{1}{{3x}}} \right)^3} = {4^3} - {\left( {\dfrac{1}{{3x}}} \right)^3} - 3{\left( 4 \right)^2}\left( {\dfrac{1}{{3x}}} \right) + 3\left( 4 \right){\left( {\dfrac{1}{{3x}}} \right)^2}$
Now simplify the above equation we have,
$ \Rightarrow {\left( {4 - \dfrac{1}{{3x}}} \right)^3} = 64 - \dfrac{1}{{27{x^3}}} - \dfrac{{16}}{x} + \dfrac{4}{{3{x^2}}}$
So, this is the required cube of the given binomial expression.

Note – Whenever we face such types of problems the key concept is to have the basic understanding of the direct algebraic formula for ${\left( {a - b} \right)^3}$. The gist of direct algebraic formula helps in direct simplification of the given problem statement.