Let's solve the expression step-by-step:
### Simplifying the expression:
[tex]\[ \frac{(4g^3h^2k^4)^3}{8g^3h^2} - (h^5k^3)^5 \][/tex]
Step 1: Simplify the fraction numerator
[tex]\[ (4g^3h^2k^4)^3 \][/tex]
First, apply the exponent:
[tex]\[ (4g^3h^2k^4)^3 = 4^3 (g^3)^3 (h^2)^3 (k^4)^3 \][/tex]
[tex]\[ = 64g^9h^6k^{12} \][/tex]
Step 2: Simplify the denominator
[tex]\[ 8g^3h^2 \][/tex]
Step 3: Simplify the division
[tex]\[ \frac{64g^9h^6k^{12}}{8g^3h^2} \][/tex]
Dividing each term separately:
[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]\[ \frac{k^{12}}{k^0} = k^{12} \][/tex]
Putting it all together:
[tex]\[ \frac{64g^9h^6k^{12}}{8g^3h^2} = 8g^6h^4k^{12} \][/tex]
Step 4: Simplify the second term
[tex]\[ (h^5k^3)^5 \][/tex]
Applying the exponent:
[tex]\[ (h^5)^5 (k^3)^5 \][/tex]
[tex]\[ = h^{5 \cdot 5} k^{3 \cdot 5} = h^{25} k^{15} \][/tex]
Step 5: Subtract the second term from the simplified fraction
[tex]\[ 8g^6h^4k^{12} - h^{25}k^{15} \][/tex]
Finally, we compare this simplified expression to the given options:
### Checking against the given options:
- [tex]\(8g^2h^3k^7 - h^{10}k^8\)[/tex]
- [tex]\(8g^9h^7k^7 - h^{10}k^8\)[/tex]
- [tex]\(8g^3h^3k^{12} - h^{25}k^{15}\)[/tex]
- [tex]\(8g^6h^4k^{12} - h^{25}k^{15}\)[/tex]
The simplified expression [tex]\(8g^6h^4k^{12} - h^{25}k^{15}\)[/tex] matches exactly with the fourth option.
Therefore, the correct expression is:
[tex]\[ 8g^6h^4k^{12} - h^{25}k^{15} \][/tex]
The correct answer is:
[tex]\[ \boxed{4} \][/tex]
