To determine which expression is equivalent to \(\frac{(2g^5)^3}{(4h^2)^3}\), we need to simplify the given expression step-by-step.
First, let's simplify the numerator \((2g^5)^3\):
[tex]\[
(2g^5)^3 = 2^3 \cdot (g^5)^3
\][/tex]
Calculate each part separately:
[tex]\[
2^3 = 8
\][/tex]
[tex]\[
(g^5)^3 = g^{5 \cdot 3} = g^{15}
\][/tex]
So the numerator becomes:
[tex]\[
8g^{15}
\][/tex]
Next, let's simplify the denominator \((4h^2)^3\):
[tex]\[
(4h^2)^3 = 4^3 \cdot (h^2)^3
\][/tex]
Calculate each part separately:
[tex]\[
4^3 = 64
\][/tex]
[tex]\[
(h^2)^3 = h^{2 \cdot 3} = h^6
\][/tex]
So the denominator becomes:
[tex]\[
64h^6
\][/tex]
Now, our expression looks like this:
[tex]\[
\frac{8g^{15}}{64h^6}
\][/tex]
We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 8:
[tex]\[
\frac{8g^{15}}{64h^6} = \frac{8 \cdot g^{15}}{8 \cdot 8h^6} = \frac{g^{15}}{8h^6}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\frac{g^{15}}{8h^6}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{g^{15}}{8h^6}}
\][/tex]
