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
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
Solution and Explanation
(i) (–12)3 + (7)3 + (5)3
Let x = -12, y = 7, and z = 5
It can be observed that, x + y + z = -12 + 7 + 5 = 0
It is known that if x + y + z = 0, then
x3 + y3 + z3 = 3xyz
∴ (-12)3 + (7)3 + (5)3 = 3(-12) (7) (5) = -1260
(ii) (28)3 + (–15)3 + (–13)3
Let x = 28, y = -15, and z = -13
It can be observed that,
x + y + z = 28 + (-15) + (-13) = 28 - 28 = 0
It is known that if x + y + z = 0,
then x3 + y3 + z3 = 3xyz
∴ (28)3 + (-15)3 + (-13)3 = 3(28) (-15) (-13)=16380
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