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

We know that,
$$x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$$

Also, if x + y + z = 0 then, $$x^3 + y^3 + z^3 = 3xyz$$

i) Here, -12 + 7 + 5 = 0

$$\therefore (-12)^3 + (5)^3 + (7))^3$$
$$= 3(-12)(7)(5)$$
$$= -1260$$

ii)Here, 28 - 15 - 13 = 0
$$\therefore (28)^3 + (-15)^3 + (-13))^3$$
$$= 3(-13)(28)(-15)$$
$$= 16380$$