\frac{1}{4}\left(27e^{\frac{8}{9}}+27e^{\frac{8}{11}}+27e^{\frac{8}{13}}+27e^{\frac{8}{15}}\right)

The given expression is:

\[
\frac{1}{4} \left(27e^{\frac{8}{9}} + 27e^{\frac{8}{11}} + 27e^{\frac{8}{13}} + 27e^{\frac{8}{15}}\right)
\]

To simplify this, we can factor out the common term \( 27 \) from the sum inside the parentheses:

\[
\frac{1}{4} \cdot 27 \left(e^{\frac{8}{9}} + e^{\frac{8}{11}} + e^{\frac{8}{13}} + e^{\frac{8}{15}}\right)
\]

Next, simplify the constant multiplication:

\[
\frac{27}{4} \left(e^{\frac{8}{9}} + e^{\frac{8}{11}} + e^{\frac{8}{13}} + e^{\frac{8}{15}}\right)
\]

This is the simplified form of the expression. If you need a numerical approximation, let me know and I can compute it.

Would you like further details or have any questions? 

Here are five related questions to expand on this:

1. What are the properties of exponential functions, and how do they behave as the exponent grows larger?
2. How can we numerically approximate this expression using a calculator?
3. What are some applications of such exponential expressions in physics or finance?
4. How would the simplification change if one of the exponents was negative?
5. How does factoring out constants help in simplifying complex expressions?

**Tip:** Factoring common terms early on in expressions can significantly simplify computations later on.

Explore how to simplify the expression \frac{1}{4}\left(27e^{\frac{8}{9}}+27e^{\frac{8}{11}}+27e^{\frac{8}{13}}+27e^{\frac{8}{15}}\right) by factoring out constants and applying exponential function properties. Suitable for students in Grades 11-12 studying exponential functions and algebraic techniques.

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