To determine which expression is equivalent to [tex]\(\frac{(2 g^5)^3}{(4 h^2)^3}\)[/tex], we can simplify it step by step:
1. Simplify the numerator: [tex]\((2 g^5)^3\)[/tex]
- Raise the constant 2 to the power of 3: [tex]\(2^3 = 8\)[/tex].
- Raise [tex]\(g^5\)[/tex] to the power of 3: [tex]\((g^5)^3 = g^{5 \cdot 3} = g^{15}\)[/tex].
So, the numerator [tex]\((2 g^5)^3\)[/tex] simplifies to [tex]\(8 g^{15}\)[/tex].
2. Simplify the denominator: [tex]\((4 h^2)^3\)[/tex]
- Express the 4 as [tex]\(2^2\)[/tex], so rewrite [tex]\(4 h^2\)[/tex] as [tex]\((2^2 h^2)\)[/tex].
- Raise [tex]\(2^2\)[/tex] to the power of 3: [tex]\((2^2)^3 = 2^{2 \cdot 3} = 2^6 = 64\)[/tex].
- Raise [tex]\(h^2\)[/tex] to the power of 3: [tex]\((h^2)^3 = h^{2 \cdot 3} = h^6\)[/tex].
So, the denominator [tex]\((4 h^2)^3\)[/tex] simplifies to [tex]\(64 h^6\)[/tex].
3. Combine the simplified numerator and denominator:
[tex]\[
\frac{(2 g^5)^3}{(4 h^2)^3} = \frac{8 g^{15}}{64 h^6}
\][/tex]
4. Simplify the fraction:
- Simplify the constant fraction: [tex]\(\frac{8}{64} = \frac{1}{8}\)[/tex].
Thus, the simplified expression is:
[tex]\[
\frac{8 g^{15}}{64 h^6} = \frac{1}{8} \cdot \frac{g^{15}}{h^6} = \frac{g^{15}}{8 h^6}
\][/tex]
Therefore, the equivalent expression is [tex]\(\frac{g^{15}}{8 h^6}\)[/tex], which matches the first choice:
[tex]\[\frac{g^{15}}{8 h^6}\][/tex]
