Question

`a + 2c + 3b = 0` then what is `a ^3 + 8 b ^3 + 27c^3`?

Answer 1

We can rewrite one of the letters as an expression in the others.

`a+2c+3b=0->a=-2c-3b`

Then we replace every `a` in the second expression:

`a^3+8b^3+27c^3=`
`(-2c-3b)^3+8b^3+27c^3=`
`(-8c^3-36bc^2-54b^2c-19b^3)+8b^3+27c^3=`

`19c^3-36bc^2-54b^2c-10b^3`

Extra :
If the first expression had been `a+2b+3c=0`
(and I suspect that it was!!)

Then the `b^3`'s and `c^3`'s would have cancelled out nicer.

`a=-2b-3c`
`(-2b-3c)^3+8b^3+27c^3=`
`(-8b^3-36b^2c-54bc^2-27c^3)+8b^3+27c^3=`

`-36b^2c-54bc^2 or 18bc(-2b-3c)`
and, since `a=-2b-3c` we can rewrite that:

Answer:
`a^3+8b^3+27c^3=18abc`