To simplify the expression [tex]\( e^{8-4 \ln (x) + 3 \ln (y)} \)[/tex], we can apply some properties of logarithms and exponents. Here's a step-by-step solution:
1. Identify and Separate the Terms:
The expression inside the exponent is [tex]\( 8 - 4 \ln(x) + 3 \ln(y) \)[/tex].
2. Distribute the Exponentiation:
We can rewrite the expression inside the exponential function for easier handling:
[tex]\[
8 - 4 \ln(x) + 3 \ln(y)
\][/tex]
3. Exponentiation and Properties of Logarithms:
We know from properties of logarithms that [tex]\(\ln(a^b) = b \ln(a)\)[/tex]. We can use this property to separate the terms in the exponent.
4. Separate the Terms inside the Exponent:
By splitting the exponentiation of the sum:
[tex]\[
e^{8 - 4 \ln(x) + 3 \ln(y)} = e^8 \cdot e^{-4 \ln(x)} \cdot e^{3 \ln(y)}
\][/tex]
5. Simplify Using [tex]\( e^{\ln(a)} = a \)[/tex]:
Apply the exponential and logarithmic properties:
[tex]\[
e^{-4 \ln(x)} = (e^{\ln(x)})^{-4} = x^{-4}
\][/tex]
[tex]\[
e^{3 \ln(y)} = (e^{\ln(y)})^3 = y^3
\][/tex]
6. Combine the Expressions:
Putting it all together:
[tex]\[
e^{8 - 4 \ln(x) + 3 \ln(y)} = e^8 \cdot x^{-4} \cdot y^3
\][/tex]
7. Rewriting the Terms:
Combine the constants and appropriate terms to get the final expression:
[tex]\[
e^{8 - 4 \ln(x) + 3 \ln(y)} = \frac{y^3 \cdot e^8}{x^4}
\][/tex]
Thus, the simplified form of the expression [tex]\( e^{8-4\ln(x)+3\ln(y)} \)[/tex] is:
[tex]\[
\boxed{\frac{y^3 e^8}{x^4}}
\][/tex]
