x^2/3= 8x. Find x?

Cube both sides:
x^2 = 512x^3

Divide both sides by x^2
1 = 512 x

Divide both sides by 512
x = 1 / 512

x^(2/3) - 8 x = 0
x^(2/3) (1 - 8x^(1/3) ) = 0
x = 0 , 8x^(1/3) = 1
x = 0 , x^(1/3) = 1 / 8
x = 0 , x = 1 / 512

x^(2/3) = 8x
3√x^2 = 8x
x^2 = (8x)^3
x^2 - 512x^3 = 0
x^2(1 - 512x) = 0

x^2 = 0
x = 0

1 - 512x = 0
512x = 1
x = 1/512

∴ x = 0 , 1/512

x^(2/3) = 8 x
x^(2/3)/x = 8
x^(2/3 – 1) = 8
x^(– 1/3) = 8
1/³√x = 8
[1/³√x]³ = (8)³
1/x = 512
x = 1 / 512

x^2/3 = 8x

Cube each side:

x^2 = 512x^3

512x^3 - x^2 = 0

x^2(512x - 1) = 0

x^2 = 0 or x = 1/512.

So, x = 0, or x = 1/512.

Also x=0

Please be clearer about the order of operations
x^2/3 can mean (x^2)/3 or x^(2/3)

If you meant x^(2/3), then the answer is as follows:
x^(2/3) = 8x
x^(2/3) - 8x = 0
( x^(2/3) )( 1 - 8x^(1/3) ) = 0
so either x^(2/3) = 0
or (1-8x^(1/3) ) = 0

which means:
x^(2/3) = 0
x = 0
--or--
1 - 8x^(1/3) = 0
1/8 = x^(1/3)
(1/8)^3 = ( x^(1/3) )^3
1/512 = x

so x = 1/512 or x = 0

x^2/3= 8x
(x^2/3)/x=8
x^(2/3-1)=8
x^(-1/3)=8
(x^(-1/3))^3=8^3
x^(-1)=512
1/x=512
x=1/512

ANSWER