When asked to factor 0.064a^3+8y^9, the first things I notice are the addition and the a^3 -- a cube.
Then "Oh, look, 8 is a cube and, so is y^9".
(2^3=8 and (y^3)^3 = y^9)
So, I'm thinking maybe it's a sum of two cubes.
But what about that 0.064?
Well, 3 right of the decimal is 1000ths and that's dividing by 1,000 which is 10 ^3, that is: color(white)(64)/1000 = color(white)(64)/10^3
OK, so 0.064 = 64/100 = 64/10^3
So all I'm left wondering about is the 64
I cross my fingers and start trying numbers.
2^3 is only 8,
3^3 is an odd number, it can't be 64
4^3 = 4*16=64 and there it is:
0.064a^3+8y^9 = color(red) ((4/10a)^3) + (2y^3)^3
Use the rule I've memorized:
color(red)(u^3)+v^3 = ( color(red)(u)+v) (color(red)(u^2) - color(red)(u) v+v^2) to get:
color(red)((4/10a)^3) + (2y^3)^3 = [ color(red)((4/10a)) + (2y^3)] [ color(red)((4/10a)^2) + color(red)((4/10a))(2y^3) +(2y^3)^2]
= [ (4/10a) + (2y^3)] [ (16/100 a^2) + (4/10a)(2y^3) +(4y^6)]
= [ 0.4a + 2y^3] [ 0.16 a^2 + 0.4 a*2y^3 +4y^6]
= [ 0.4a + 2y^3] [ 0.16 a^2 + 0.8 ay^3 +4y^6].
