Question

Without actually calculating the cubes, find the value of the following: ${\left(-12\right)}^3+{\left(7\right)}^3+{\left(5\right)}^3$

Answer 1

Find the value of ${\left(-12\right)}^3+{\left(7\right)}^3+{\left(5\right)}^3$.

We know, ${\mathbf{a}}^{\mathbf{3}}\mathbf{+}{\mathbf{b}}^{\mathbf{3}}\mathbf{+}{\mathbf{c}}^{\mathbf{3}}\mathbf{-}\mathbf{3}\mathbf{abc}\mathbf{=}\left(a+b+c\right)\left(a^2+b^2+c^2-\mathrm{ab}-\mathrm{bc}-\mathrm{ca}\right)$

And, ${\mathbf{a}}^{\mathbf{3}}\mathbf{+}{\mathbf{b}}^{\mathbf{3}}\mathbf{+}{\mathbf{c}}^{\mathbf{3}}\mathbf{=}\mathbf{3}\mathbf{abc}$ if $\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{+}\mathbf{c}\mathbf{=}\mathbf{0}$.

Now, ${\left(-12\right)}^3+{\left(7\right)}^3+{\left(5\right)}^3$

Here, $a=-12,b=7,c=5$

The sum is

$$\begin{gathered} =\left(-12\right)+7+5 \\ -12+12 \\ =0 \end{gathered}$$

Now,

$$\begin{gathered} {\left(-12\right)}^3+{\left(7\right)}^3+{\left(5\right)}^3=3\times -12\times 7\times 5 \\ =-1260 \end{gathered}$$

Thus, the value of ${\left(-12\right)}^3+{\left(7\right)}^3+{\left(5\right)}^3=-1260$.