To solve 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 need to simplify each part step-by-step.
## Simplify the numerator [tex]\((4 g^3 h^2 k^4)^3\)[/tex]:
The expression inside the parentheses is raised to the power of 3:
[tex]\[
(4 g^3 h^2 k^4)^3 = (4)^3 (g^3)^3 (h^2)^3 (k^4)^3
\][/tex]
Applying the exponents:
[tex]\[
(4)^3 = 64
\][/tex]
[tex]\[
(g^3)^3 = g^{3 \cdot 3} = g^9
\][/tex]
[tex]\[
(h^2)^3 = h^{2 \cdot 3} = h^6
\][/tex]
[tex]\[
(k^4)^3 = k^{4 \cdot 3} = k^{12}
\][/tex]
So the expression is now:
[tex]\[
64 g^9 h^6 k^{12}
\][/tex]
## Divide by [tex]\(8 g^3 h^2\)[/tex]:
Now we need to divide the result by [tex]\(8 g^3 h^2\)[/tex]:
[tex]\[
\frac{64 g^9 h^6 k^{12}}{8 g^3 h^2}
\][/tex]
Simplify the coefficients:
[tex]\[
\frac{64}{8} = 8
\][/tex]
Simplify the variables by subtracting the exponents of the same base:
[tex]\[
g^{9 - 3} = g^6
\][/tex]
[tex]\[
h^{6 - 2} = h^4
\][/tex]
[tex]\[
k^{12}\ \text{remains as it is, since there is no \(k\) in the denominator}
\][/tex]
Thus, we get:
[tex]\[
8 g^6 h^4 k^{12}
\][/tex]
## Simplify the second term [tex]\((h^5 k^3)^5\)[/tex]:
Raise each part inside the parentheses to the power of 5:
[tex]\[
(h^5 k^3)^5 = (h^5)^5 (k^3)^5
\][/tex]
Applying the exponents:
[tex]\[
(h^5)^5 = h^{5 \cdot 5} = h^{25}
\][/tex]
[tex]\[
(k^3)^5 = k^{3 \cdot 5} = k^{15}
\][/tex]
Thus, the second term simplifies to:
[tex]\[
h^{25} k^{15}
\][/tex]
## Combine and subtract the expressions:
Now we combine the results:
[tex]\[
8 g^6 h^4 k^{12} - h^{25} k^{15}
\][/tex]
Finally, the expression equivalent to [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] is:
[tex]\[
8 g^6 h^4 k^{12} - h^{25} k^{15}
\][/tex]
Among the given options, this matches exactly with:
[tex]\[
8 g^6 h^4 k^{12} - h^{25} k^{15}
\][/tex]
Therefore, the correct answer is:
[tex]\[
8 g^6 h^4 k^{12} - h^{25} k^{15}
\][/tex]
