Without actually calculating the cubes, find the value of each of the following:
(i) $ (-12)^{3}+(7)^{3}+(5)^{3} $
(ii) $ (28)^{3}+(-15)^{3}+(-13)^{3} $

To do:

We have to find the values of each of the given expressions.

Solution:

$x^{3}+y^{3}+z^{3}-3xyz=(x+y+z)(x^{2}+y^{2}+z^{2}-xy-yz-zx)$

If $x+y+z =0$, then

$x^{3}+y^{3}+z^{3}-3xyz= (0)(x^{2}+y^{2}+z^{2}-xy-yz-zx)$

$x^{3}+y^{3}+z^{3}-3xyz=0$

$x^{3}+y^{3}+z^{3}=3xyz$ 

(i) \( (-12)^{3}+(7)^{3}+(5)^{3} \)

Here,

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

This implies,

$a+b+c=-12+7+5$

$=-12+12$

$=0$

Therefore,

$(-12)^{3}+(7)^{3}+(5)^{3}=3(-12)(7)(5)$

$=-36\times35$

$=-1260$

(ii) \( (28)^{3}+(-15)^{3}+(-13)^{3} \)

$a=28, b=-15, c=-13$

This implies,

$a+b+c=28+(-15)+(-13)$

$=28-(15+13)$

$=28-28$

$=0$

Therefore,

$(28)^{3}+(-15)^{3}+(-13)^{3}=3(28)(-15)(-13)$

$=84\times195$

$=16380$

Tutorialspoint

Updated on: 10-Oct-2022

32 Views

Kickstart Your Career

Get certified by completing the course

Get Started