You have the equation:

\[
x^{25} - 0.2 = 500
\]

To solve for \( x \), follow these steps:

1. **Isolate the term with \( x \)**:
   \[
   x^{25} = 500 + 0.2
   \]
   \[
   x^{25} = 500.2
   \]

2. **Take the 25th root of both sides**:
   \[
   x = \sqrt[25]{500.2}
   \]

3. **Calculate the 25th root**:
   \[
   x \approx 1.8138
   \]

So, the solution is approximately \( x \approx 1.8138 \).

Would you like more details or have any other questions? Here are some related questions you might find interesting:

1. How would the solution change if the exponent were different, say \( x^{10} - 0.2 = 500 \)?
2. What if the equation were \( x^{25} + 0.2 = 500 \)?
3. How do we solve the equation if it involves a negative exponent, like \( x^{-25} - 0.2 = 500 \)?
4. How do you calculate roots for very large or very small numbers?
5. How would you solve the equation \( x^{25} = 0 \)?

**Tip:** When dealing with roots of high degree, using a calculator or computational tool can help ensure accuracy in your solution.

Learn how to solve the equation x^25 - 0.2 = 500 with detailed steps and calculations. This problem involves exponential equations and finding roots of numbers, applicable for advanced high school students.

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