Let's start by simplifying the given expression step by step:
[tex]\[
\frac{(2 g^5)^3}{(4 h^2)^3}
\][/tex]
1. Simplify the numerator [tex]\((2 g^5)^3\)[/tex]:
To simplify [tex]\((2 g^5)^3\)[/tex], apply the power of a product rule [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex]:
[tex]\[
(2 g^5)^3 = 2^3 \cdot (g^5)^3
\][/tex]
Calculate each part:
[tex]\[
2^3 = 8
\][/tex]
[tex]\[
(g^5)^3 = g^{5 \cdot 3} = g^{15}
\][/tex]
Therefore,
[tex]\[
(2 g^5)^3 = 8 g^{15}
\][/tex]
2. Simplify the denominator [tex]\((4 h^2)^3\)[/tex]:
To simplify [tex]\((4 h^2)^3\)[/tex], apply the power of a product rule [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex]:
[tex]\[
(4 h^2)^3 = 4^3 \cdot (h^2)^3
\][/tex]
Calculate each part:
[tex]\[
4^3 = 64
\][/tex]
[tex]\[
(h^2)^3 = h^{2 \cdot 3} = h^6
\][/tex]
Therefore,
[tex]\[
(4 h^2)^3 = 64 h^6
\][/tex]
3. Combine the simplified numerator and denominator:
Now we can form the fraction with the simplified numerator and denominator:
[tex]\[
\frac{8 g^{15}}{64 h^6}
\][/tex]
4. Simplify the fraction:
Divide the numerator and the denominator by their greatest common divisor, which is 8:
[tex]\[
\frac{8 g^{15}}{64 h^6} = \frac{8}{64} \cdot \frac{g^{15}}{h^6} = \frac{1}{8} \cdot \frac{g^{15}}{h^6} = \frac{g^{15}}{8 h^6}
\][/tex]
Hence, the expression [tex]\(\frac{(2 g^5)^3}{(4 h^2)^3}\)[/tex] simplifies to:
[tex]\[
\frac{g^{15}}{8 h^6}
\][/tex]
So, the equivalent expression is [tex]\(\boxed{\frac{g^{15}}{8 h^6}}\)[/tex].
