Answer :
To find the equivalent expression for \(\frac{\left(4 g^3 h^2 k^4\right)^3}{8 g^3 h^2} - \left(h^5 k^3\right)^5\), we need to first simplify each part independently.
### Step 1: Simplify the first part
We start with \(\frac{\left(4 g^3 h^2 k^4\right)^3}{8 g^3 h^2}\).
1. Calculate the numerator \(\left(4 g^3 h^2 k^4\right)^3\):
[tex]\[ \left(4 g^3 h^2 k^4\right)^3 = 4^3 \cdot (g^3)^3 \cdot (h^2)^3 \cdot (k^4)^3 = 64 \cdot g^9 \cdot h^6 \cdot k^{12} \][/tex]
2. Calculate the denominator:
[tex]\[ 8 g^3 h^2 \][/tex]
3. Divide the numerator by the denominator:
[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:
[tex]\[ g^9 / g^3 = g^{9-3} = g^6 \][/tex]
[tex]\[ h^6 / h^2 = h^{6-2} = h^4 \][/tex]
[tex]\[ k^{12} \text{ remains as } k^{12} \text{ since there is no } k \text{ in the denominator} \][/tex]
Combining these, we get:
[tex]\[ 8 g^6 h^4 k^{12} \][/tex]
### Step 2: Simplify the second part
We move on to \(\left(h^5 k^3\right)^5\):
[tex]\[ \left(h^5 k^3\right)^5 = (h^5)^5 \cdot (k^3)^5 = h^{25} \cdot k^{15} \][/tex]
### Step 3: Subtract the two parts
Now, we need to subtract the second part from the first part:
[tex]\[ 8 g^6 h^4 k^{12} - h^{25} k^{15} \][/tex]
### Conclusion
The fully simplified equivalent expression is:
[tex]\[ 8 g^6 h^4 k^{12} - h^{25} k^{15} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{8 g^6 h^4 k^{12} - h^{25} k^{15}} \][/tex]
So, the corresponding option is:
[tex]\[ \boxed{4} \][/tex]
### Step 1: Simplify the first part
We start with \(\frac{\left(4 g^3 h^2 k^4\right)^3}{8 g^3 h^2}\).
1. Calculate the numerator \(\left(4 g^3 h^2 k^4\right)^3\):
[tex]\[ \left(4 g^3 h^2 k^4\right)^3 = 4^3 \cdot (g^3)^3 \cdot (h^2)^3 \cdot (k^4)^3 = 64 \cdot g^9 \cdot h^6 \cdot k^{12} \][/tex]
2. Calculate the denominator:
[tex]\[ 8 g^3 h^2 \][/tex]
3. Divide the numerator by the denominator:
[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:
[tex]\[ g^9 / g^3 = g^{9-3} = g^6 \][/tex]
[tex]\[ h^6 / h^2 = h^{6-2} = h^4 \][/tex]
[tex]\[ k^{12} \text{ remains as } k^{12} \text{ since there is no } k \text{ in the denominator} \][/tex]
Combining these, we get:
[tex]\[ 8 g^6 h^4 k^{12} \][/tex]
### Step 2: Simplify the second part
We move on to \(\left(h^5 k^3\right)^5\):
[tex]\[ \left(h^5 k^3\right)^5 = (h^5)^5 \cdot (k^3)^5 = h^{25} \cdot k^{15} \][/tex]
### Step 3: Subtract the two parts
Now, we need to subtract the second part from the first part:
[tex]\[ 8 g^6 h^4 k^{12} - h^{25} k^{15} \][/tex]
### Conclusion
The fully simplified equivalent expression is:
[tex]\[ 8 g^6 h^4 k^{12} - h^{25} k^{15} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{8 g^6 h^4 k^{12} - h^{25} k^{15}} \][/tex]
So, the corresponding option is:
[tex]\[ \boxed{4} \][/tex]
