To simplify the given expression [tex]\(\frac{\left(4 g^3 h^2 k^4\right)^3}{8 g^3 h^2} - \left(h^5 k^3\right)^5\)[/tex], we will break it down step-by-step:
1. Simplify [tex]\((4 g^3 h^2 k^4)^3\)[/tex]:
[tex]\[
(4 g^3 h^2 k^4)^3 = (4)^3 \cdot (g^3)^3 \cdot (h^2)^3 \cdot (k^4)^3
\][/tex]
Calculate each term:
[tex]\[
4^3 = 64
\][/tex]
[tex]\[
(g^3)^3 = g^{3 \times 3} = g^9
\][/tex]
[tex]\[
(h^2)^3 = h^{2 \times 3} = h^6
\][/tex]
[tex]\[
(k^4)^3 = k^{4 \times 3} = k^{12}
\][/tex]
So, the expression becomes:
[tex]\[
(4 g^3 h^2 k^4)^3 = 64 g^9 h^6 k^{12}
\][/tex]
2. Simplify the fraction [tex]\(\frac{64 g^9 h^6 k^{12}}{8 g^3 h^2}\)[/tex]:
Separate the fraction into individual terms:
[tex]\[
\frac{64}{8} \cdot \frac{g^9}{g^3} \cdot \frac{h^6}{h^2} \cdot k^{12}
\][/tex]
Simplify each fraction:
[tex]\[
\frac{64}{8} = 8
\][/tex]
[tex]\[
\frac{g^9}{g^3} = g^{9-3} = g^6
\][/tex]
[tex]\[
\frac{h^6}{h^2} = h^{6-2} = h^4
\][/tex]
[tex]\[
k^{12} \text{ remains as } k^{12}
\][/tex]
So, the simplified fraction is:
[tex]\[
\frac{64 g^9 h^6 k^{12}}{8 g^3 h^2} = 8 g^6 h^4 k^{12}
\][/tex]
3. Simplify [tex]\((h^5 k^3)^5\)[/tex]:
[tex]\[
(h^5 k^3)^5 = (h^5)^5 \cdot (k^3)^5
\][/tex]
Calculate each term:
[tex]\[
(h^5)^5 = h^{5 \times 5} = h^{25}
\][/tex]
[tex]\[
(k^3)^5 = k^{3 \times 5} = k^{15}
\][/tex]
So, the expression becomes:
[tex]\[
(h^5 k^3)^5 = h^{25} k^{15}
\][/tex]
4. Combine the simplified terms:
[tex]\[
8 g^6 h^4 k^{12} - h^{25} k^{15}
\][/tex]
Therefore, the equivalent expression is:
[tex]\[
8 g^6 h^4 k^{12} - h^{25} k^{15}
\][/tex]
So, the correct option is:
[tex]\[
\boxed{8 g^6 h^4 k^{12} - h^{25} k^{15}}
\][/tex]
