The equation presented is:

$4^{4 \log 5} + 3^{3 \log 4}$

To simplify this expression, let's use the property of logarithms and exponents, which states that:

$a^{b \log c} = c^{\log a^b}$

Simplification of $4^{4 \log 5}$:

Using the property $a^{b \log c} = c^{\log a^b}$, we have:

$4^{4 \log 5} = 5^{\log 4^4} = 5^{\log 256}$

Simplification of $3^{3 \log 4}$:

Similarly, applying the same property:

$3^{3 \log 4} = 4^{\log 3^3} = 4^{\log 27}$

Now the expression becomes:

$5^{\log 256} + 4^{\log 27}$

At this point, solving it numerically:

1. $\log 256 \approx 2.408$, so $5^{2.408} \approx 393.66$
2. $\log 27 \approx 1.431$, so $4^{1.431} \approx 10.23$

Adding the results:

$393.66 + 10.23 = 403.89$

So, the final answer is approximately 404, which corresponds to option E.9 on the provided list.

Would you like more details or have any questions? Here are some related questions:

1. How can you simplify exponential expressions involving logarithms?
2. What other logarithmic properties can simplify calculations?
3. How do you convert between logarithmic and exponential forms?
4. Can you think of other real-world applications of logarithmic and exponential functions?
5. What is the difference between common logarithms and natural logarithms?

Tip: When simplifying expressions involving logarithms and exponents, always check for the possibility of applying logarithmic properties to reduce the complexity.