Expression is $x^{1/2} \times Y^{-1} \times z^{2/3}$ with $x = 9$, $Y = 3$, and $z = 8$.

Express each variable as a power of its prime factors: $9 = 3^2$ and $8 = 2^3$. This simplifies the exponents instantly: $(3^2)^{1/2} = 3^1$ and $(2^3)^{2/3} = 2^2$.

Students often struggle with negative exponents ($Y^{-1} = 1/Y$) or miscalculate the fractional exponent of $8$ as $8^2$ instead of $(8^{1/3})^2$.

This concept is fundamental for simplifying complex algebraic expressions and solving logarithmic equations where base conversion is required.