Let's simplify the given expression step-by-step:
[tex]\[ \frac{(2g^5)^3}{(4h^2)^3} \][/tex]
1. Apply the power rule: When raising a product to a power, distribute the exponent to each factor inside the parentheses.
[tex]\[ (2g^5)^3 = 2^3 \cdot (g^5)^3 \][/tex]
[tex]\[ (4h^2)^3 = 4^3 \cdot (h^2)^3 \][/tex]
2. Simplify the exponents: Raise the individual parts to the powers.
[tex]\[ 2^3 = 8 \][/tex]
[tex]\[ (g^5)^3 = g^{15} \][/tex]
[tex]\[ 4^3 = 64 \][/tex]
[tex]\[ (h^2)^3 = h^{6} \][/tex]
So, now we have:
[tex]\[ \frac{8g^{15}}{64h^6} \][/tex]
3. Simplify the constants (numbers):
[tex]\[ \frac{8}{64} = \frac{1}{8} \][/tex]
4. Combine the simplified terms:
After simplifying the constants:
[tex]\[ = \frac{g^{15}}{8h^6} \][/tex]
Therefore, the expression equivalent to [tex]\( \frac{(2g^5)^3}{(4h^2)^3} \)[/tex] is:
[tex]\[ \boxed{\frac{g^{15}}{8h^6}} \][/tex]
Hence, the correct answer is:
[tex]\[ \frac{g^{15}}{8h^6} \][/tex]
