To simplify the expression:

$2^{100000} - \frac{2}{4^{25000}} + 1$

First, let's break it down step by step.

Step 1: Simplify the power terms

We know that $4^{25000} = (2^2)^{25000} = 2^{50000}$. This allows us to rewrite the denominator:

$\frac{2}{4^{25000}} = \frac{2}{2^{50000}} = 2^{1 - 50000} = 2^{-49999}$

Step 2: Rewrite the entire expression

Now, substitute this back into the original expression:

$2^{100000} - 2^{-49999} + 1$

Step 3: Final form

This is as simplified as it gets. The final simplified expression is:

$2^{100000} - 2^{-49999} + 1$

This form is exact and cannot be simplified further unless you're looking for an approximate numerical value, which would be very large due to the power of 2 terms.

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Do you want more details or have any questions about the simplification process?

Here are 5 related questions for further exploration:

1. How would you approximate $2^{100000}$ in scientific notation?
2. What are some real-world applications of large exponent calculations like $2^{100000}$?
3. How do powers of two grow compared to other exponential functions?
4. Can negative exponents always be simplified like in this case?
5. What happens if the powers involved are irrational numbers?

Tip: When simplifying exponential expressions, try to express everything in terms of the same base (like 2 in this example). This makes simplification much easier.