Question

Without actually calculating the cubes find the value of (0.2)cube + (0.6)cube + (-0.8)cube

Answer 1

Algebra,

If we don't calculate it directly we have to use this Identitie,

a³ + b³ + c³ = (a + b + c)(a²+ b²+ c²-ab- bc- ca) + 3abc

Now we have,

(0.2)³+(0.6)³-(0.8)³

= (1/5)³+ (3/5)³- (4/5)³

= (1/5 + 3/5 - 4/5){(1/5)²+ (3/5)²+ (4/5)²-(3/25) -(-12/25) -(-4/25)} + 3(-12/125)

= [{(4-4)}/5]{(1/5)²+ (3/5)²+ (4/5)²-(3/25) +(-12/25) -(-4/25)} +36/125

= 0×{(1/5)²+ (3/5)²+ (4/5)²-(3/25) -(-12/25) -(-4/25)} - 36/125

Which is - 36/125

Quick cheak,

(0.2)³+(0.6)³-(0.8)³

= (1/5)³+ (3/5)³- (4/5)³

= 1/125 + 27/125 - 64/125

= -36/125

That's it,
Hope it helped (･ิω･ิ)