Question

What is the formula of $\left(a^3+b^3\right)$?

Answer 1

Find the formula for $\left(a^3+b^3\right)$.

Since, for any real number $a\mathrm{and}b$

${\left(a+b\right)}^{\mathbf{3}}\mathbf{=}{\mathit{a}}^{\mathbf{3}}\mathbf{+}{\mathit{b}}^{\mathbf{3}}\mathbf{+}\mathbf{3}\mathit{a}\mathit{b}\left(a+b\right)\mathbf{}\dots \left(1\right)$

From the equation$\left(1\right)$, the value $a^3+b^3$ is derived as follows:

$$\begin{gathered} \left(a^3+b^3\right)={\left(a+b\right)}^3-3ab\left(a+b\right) \\ \Rightarrow \left({\begin{array}{l}a\end{array}}^3+b^3\right)=\begin{array}{l}\left(a+b\right)\left({\left(a+b\right)}^2-3ab\right)\dots \left(2\right)\end{array} \end{gathered}$$

Substitute the value of ${\left(a+b\right)}^2$ from ${\left(a+b\right)}^{\mathbf{2}}\mathbf{=}{\mathit{a}}^{\mathbf{2}}\mathbf{+}{\mathit{b}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{a}\mathit{b}$ in $\left(2\right)$:

$\begin{array}{rcl}\left(a^3+b^3\right)& =& \left(a+b\right)\left(a^2+b^2+2ab-3ab\right)\\ & \Rightarrow & \left(a^3+b^3\right)=\left(a+b\right)\left(a^2+b^2-ab\right)\end{array}$

Hence, the formula for $\left(a^3+b^3\right)$ is $\begin{array}{rcl}\left(a^3+b^3\right)& =& \left(a+b\right)\left(a^2+b^2-ab\right)\end{array}$.