To find the expression equivalent to [tex]\(\frac{(2 g^5)^3}{(4 h^2)^3}\)[/tex], let's solve this step-by-step.
Step 1: Apply the power rule to the entire numerator and denominator.
The power rule states [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex].
For the numerator [tex]\((2 g^5)^3\)[/tex]:
[tex]\[
(2 g^5)^3 = 2^3 \cdot (g^5)^3
\][/tex]
Simplify by applying the power to each:
[tex]\[
2^3 = 8 \quad \text{and} \quad (g^5)^3 = g^{5 \cdot 3} = g^{15}
\][/tex]
So, the numerator simplifies to:
[tex]\[
(2 g^5)^3 = 8 g^{15}
\][/tex]
For the denominator [tex]\((4 h^2)^3\)[/tex]:
[tex]\[
(4 h^2)^3 = 4^3 \cdot (h^2)^3
\][/tex]
Simplify by applying the power to each:
[tex]\[
4^3 = 64 \quad \text{and} \quad (h^2)^3 = h^{2 \cdot 3} = h^6
\][/tex]
So, the denominator simplifies to:
[tex]\[
(4 h^2)^3 = 64 h^6
\][/tex]
Step 2: Combine the results from the numerator and denominator.
Now, the entire expression simplifies to:
[tex]\[
\frac{(2 g^5)^3}{(4 h^2)^3} = \frac{8 g^{15}}{64 h^6}
\][/tex]
Step 3: Simplify the fraction [tex]\(\frac{8}{64}\)[/tex].
The constant fraction simplifies as follows:
[tex]\[
\frac{8}{64} = \frac{1}{8}
\][/tex]
Step 4: Combine the simplified constants and variables.
So, the simplified expression is:
[tex]\[
\frac{8 g^{15}}{64 h^6} = \frac{g^{15}}{8 h^6}
\][/tex]
Conclusion:
The correct expression equivalent to [tex]\(\frac{(2 g^5)^3}{(4 h^2)^3}\)[/tex] is:
[tex]\[
\frac{g^{15}}{8 h^6}
\][/tex]
So, the correct choice is:
[tex]\[
\boxed{\frac{g^{15}}{8 h^6}}
\][/tex]
