To solve the given expression, we need to perform a detailed step-by-step simplification. The given expression is:
[tex]\[
\frac{(4 g^3 h^2 k^4)^3}{8 g^3 h^2} - (h^5 k^3)^5
\][/tex]
Let's break it down into parts:
### Step 1: Simplify the Numerator [tex]\((4 g^3 h^2 k^4)^3\)[/tex]
First, take powers of each term inside the parentheses:
[tex]\[
(4 g^3 h^2 k^4)^3 = 4^3 (g^3)^3 (h^2)^3 (k^4)^3
\][/tex]
Simplifying each part:
[tex]\[
4^3 = 64, \quad (g^3)^3 = g^9, \quad (h^2)^3 = h^6, \quad (k^4)^3 = k^{12}
\][/tex]
So the simplified form of the numerator is:
[tex]\[
64 g^9 h^6 k^{12}
\][/tex]
### Step 2: Simplify the Fraction
Now we have:
[tex]\[
\frac{64 g^9 h^6 k^{12}}{8 g^3 h^2}
\][/tex]
Perform the division for each term separately:
[tex]\[
\frac{64}{8} = 8, \quad \frac{g^9}{g^3} = g^{9-3} = g^6, \quad \frac{h^6}{h^2} = h^{6-2} = h^4, \quad \frac{k^{12}}{} = k^{12}
\][/tex]
So the simplified form of the fraction is:
[tex]\[
8 g^6 h^4 k^{12}
\][/tex]
### Step 3: Simplify the Second Term [tex]\((h^5 k^3)^5\)[/tex]
Take powers of each term inside the parentheses:
[tex]\[
(h^5 k^3)^5 = (h^5)^5 (k^3)^5
\][/tex]
Simplify each part:
[tex]\[
(h^5)^5 = h^{25}, \quad (k^3)^5 = k^{15}
\][/tex]
So the simplified form is:
[tex]\[
h^{25} k^{15}
\][/tex]
### Step 4: Combine the Terms and Perform the Subtraction
Now we have:
[tex]\[
8 g^6 h^4 k^{12} - h^{25} k^{15}
\][/tex]
This expression cannot be simplified further because the terms are not like terms. Hence, the final simplified form for the given expression is:
[tex]\[
8 g^6 h^4 k^{12} - h^{25} k^{15}
\][/tex]
Comparing this with the given options:
[tex]\[
8 g^2 h^3 k^7 - h^{10} k^8
\][/tex]
[tex]\[
8 g^9 h^7 k^7 - h^{10} k^8
\][/tex]
[tex]\[
8 g^3 h^3 k^{12} - h^{25} k^{15}
\][/tex]
[tex]\[
8 g^6 h^4 k^{12} - h^{25} k^{15}
\][/tex]
The correct answer is:
[tex]\[
8 g^6 h^4 k^{12} - h^{25} k^{15}
\][/tex]
