Question

What is the formula for $a^3-b^3$?

Answer 1

Hint – In this question use the algebraic identity of ${\left(a-b\right)}^3$ which is $a^3-b^3-3a^{2}b+3a{b}^2$. Then take terms of this to a different side leaving behind only $a^3-b^3$. Simplification of the other side will help getting the right answer for $a^3-b^3$.

Complete step-by-step answer:
As we know that ${\left(a-b\right)}^3=a^3-b^3-3a^{2}b+3a{b}^2$
$\Rightarrow a^3-b^3={\left(a-b\right)}^3+3a^{2}b-3a{b}^2$
$\Rightarrow a^3-b^3={\left(a-b\right)}^3+3ab\left(a-b\right)$
Now take (a – b) common we have,
$\Rightarrow a^3-b^3=\left(a-b\right)\left[{\left(a-b\right)}^2+3ab\right]$
Now as we know that ${\left(a-b\right)}^2=a^2+b^2-2ab$ so use this property in above equation we have,
$\Rightarrow a^3-b^3=\left(a-b\right)\left[a^2+b^2-2ab+3ab\right]$
Now simplify it we have,
$\Rightarrow a^3-b^3=\left(a-b\right)\left[a^2+b^2+ab\right]$
So this is the required formula of $a^3-b^3$.

Note – The formula for $(a-b{)}^3$ can easily be derived by writing $(a-b{)}^3=(a-b)(a-b{)}^2$. Then using the algebraic identity that $(a-b{)}^2=a^2+b^2-2ab$, we can write $(a-b{)}^3=(a-b)(a^2+b^2-2ab)$, on simplification of this right hand side will help us getting the value of$(a-b{)}^3$. So if somehow we find difficulty remembering this formula then the above steps can be used to derive it immediately.