Question

which expression is equivalent to \\(\frac{(2g^5)^3}{(4h^2)^3}\\)?\
\\(\frac{g^{15}}{8h^6}\\)\
\\(\frac{g^5}{2h^2}\\)\
\\(\frac{g^{15}}{2h^6}\\)\
\\(\frac{g^8}{8h^5}\\)

Explanation:

Step1: Apply power of a product rule

For a power of a product \((ab)^n = a^n b^n\), so we expand the numerator and the denominator:
\[
\frac{(2g^{5})^{3}}{(4h^{2})^{3}}=\frac{2^{3}(g^{5})^{3}}{4^{3}(h^{2})^{3}}
\]

Step2: Simplify the exponents

Using the power of a power rule \((a^m)^n=a^{mn}\) and calculate the numerical powers:
\(2^{3} = 8\), \(4^{3}=64\), \((g^{5})^{3}=g^{15}\), \((h^{2})^{3}=h^{6}\)
So the expression becomes:
\[
\frac{8g^{15}}{64h^{6}}
\]

Step3: Simplify the fraction of the numerical coefficients

Simplify \(\frac{8}{64}=\frac{1}{8}\)
So the final simplified expression is:
\[
\frac{g^{15}}{8h^{6}}
\]

Answer:

\(\boldsymbol{\frac{g^{15}}{8h^{6}}}\) (corresponding to the first option: \(\frac{g^{15}}{8h^{6}}\))