Without actually calculating the cubes, find the values of ${(- 15)}^{3} + 7^{3} + 8^{3}$

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Solution

We know the corollary: if $a + b + c = 0$ then $a^{3} + b^{3} + c^{3} = 3 a b c$

Using the above corollary taking $a = - 15$ , $b = 7$ and $c = 8$ , we have $a + b + c = - 15 + 7 + 8 = 0$ then the value of $(- 15)^{3} + (7)^{3} + (8)^{3}$ is:

$(- 15)^{3} + (7)^{3} + (8)^{3} = 3 (- 15 \times 7 \times 8) = - 2520$

Hence, $(- 15)^{3} + (7)^{3} + (8)^{3} = - 2520$

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