To simplify the expression \( e^{-2 \ln 5} \), we can use the properties of logarithms and exponentials. Specifically, we can use the property that \( e^{\ln x} = x \).

Here's the step-by-step simplification:

\[ e^{-2 \ln 5} \]

First, we use the property of logarithms that allows us to move the coefficient inside the logarithm as an exponent:

\[ e^{-2 \ln 5} = e^{\ln (5^{-2})} \]

Next, we apply the property \( e^{\ln x} = x \):

\[ e^{\ln (5^{-2})} = 5^{-2} \]

Finally, we simplify the exponent:

\[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \]

So, the simplified form of \( e^{-2 \ln 5} \) is:

\[ \boxed{\frac{1}{25}} \]

Would you like any further details or have any questions on this solution?

Here are 8 related questions to expand on the topic:

1. How would you simplify \( e^{3 \ln 4} \)?
2. Can you express \( e^{-\ln 3} \) in a simpler form?
3. What is the value of \( e^{2 \ln 7} \)?
4. How do you solve \( e^{\ln (x^2)} \)?
5. What is the result of \( e^{-\ln 9} \)?
6. How do you simplify \( e^{\ln (x^3)} \)?
7. Can you calculate \( e^{4 \ln 2} \)?
8. How would you solve \( e^{-3 \ln 10} \)?

**Tip:** When dealing with exponentials and logarithms, try to use the properties of exponents and logarithms to simplify expressions step by step.

Learn how to simplify the expression e^(-2 ln 5) using properties of exponentials and logarithms. This detailed step-by-step solution covers moving coefficients inside logarithms as exponents and simplifying using exponential properties. Suitable for advanced high school students and anyone interested in understanding exponentials and logarithms.

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