Question

Find the value of ${\left( {48} \right)^3} - {\left( {30} \right)^3} - {\left( {18} \right)^3}$.

Answer 1

Hint – In this question express 48 as the sum of 30+18. Then it gets in the form an algebraic identity${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}$. Simplify this expression to get the answer.

Complete step-by-step answer:

Given equation is
${\left( {48} \right)^3} - {\left( {30} \right)^3} - {\left( {18} \right)^3}$

As we know (48 = 30 + 18) so use this we can written above equation as
$ \Rightarrow {\left( {30 + 18} \right)^3} - {\left( {30} \right)^3} - {\left( {18} \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 expand the cube in above equation we have,
$ \Rightarrow {\left( {30} \right)^3} + {\left( {18} \right)^3} + 3{\left( {30} \right)^2}\left( {18} \right) + 3\left( {30} \right){\left( {18} \right)^2} - {\left( {30} \right)^3} - {\left( {18} \right)^3}$

Now cancel out the terms we have,
$ \Rightarrow 3{\left( {30} \right)^2}\left( {18} \right) + 3\left( {30} \right){\left( {18} \right)^2}$

Now take common terms as common we have,
$ \Rightarrow 3\left( {30} \right)\left( {18} \right)\left( {30 + 18} \right)$
$ \Rightarrow 3\left( {30} \right)\left( {18} \right)\left( {48} \right)$

Now simply multiply these numbers we have,
= 77,760

So this is the required answer.

Note – There can be another method to solve this problem which is based on direct calculation of the cubes of the given number and then performing the simplification as required in question, but it will be lengthy and would take a lot of time so the method above mentioned is preferred.