Answer: \(x=3\).

Write every term as a power of \(2\):

\(4^{x+1}=2^{2x+2},\qquad 32^{x/3}=2^{5x/3},\qquad 8^{1/3}=2.\)

Then

\(\frac{4^{x+1}}{2^{x-1}}=\frac{2^{2x+2}}{2^{x-1}}=2^{x+3}.\)

The right-hand side is

\(2^{5x/3}\times2=2^{5x/3+1}.\)

Therefore

\(2^{x+3}=2^{5x/3+1}.\)

Equate the powers:

\(x+3=\frac{5x}{3}+1.\)

So

\(2=\frac{2x}{3},\)

and hence

\(x=3.\)