https://file.aiquickdraw.com/mathbot/file/1723329353591aykrou8i.jpg

This image shows a nested radical expression:

\[
\sqrt{5 + \sqrt{100 + \sqrt{160 + \sqrt{500 + \sqrt{25}}}}}
\]

Let's solve this step by step:

1. **Start with the innermost radical:**
   \[
   \sqrt{25} = 5
   \]

2. **Move to the next radical:**
   \[
   \sqrt{500 + 5} = \sqrt{505}
   \]

   \[
   \sqrt{505} \approx 22.47
   \]

3. **Next radical:**
   \[
   \sqrt{160 + 22.47} = \sqrt{182.47}
   \]

   \[
   \sqrt{182.47} \approx 13.51
   \]

4. **Next radical:**
   \[
   \sqrt{100 + 13.51} = \sqrt{113.51}
   \]

   \[
   \sqrt{113.51} \approx 10.65
   \]

5. **Final radical:**
   \[
   \sqrt{5 + 10.65} = \sqrt{15.65}
   \]

   \[
   \sqrt{15.65} \approx 3.96
   \]

So, the value of the entire nested radical expression is approximately **3.96**.

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

1. What is the value of \(\sqrt{625}\)?
2. How does the process of solving nested radicals differ from simple square roots?
3. Solve \(\sqrt{100 + \sqrt{25}}\).
4. What are the steps to simplify a nested radical expression like \(\sqrt{a + \sqrt{b}}\)?
5. Can you simplify \(\sqrt{9 + \sqrt{36 + \sqrt{64}}}\)?
6. How does approximation affect the final result in nested radical expressions?
7. What methods can be used to simplify nested radicals by hand?
8. What is the relationship between nested radicals and iterative functions?

**Tip:** When dealing with nested radicals, always start by simplifying the innermost radical first.

Learn how to solve a nested radical expression step-by-step with detailed explanations. This problem involves nested square roots and requires approximation techniques. Suitable for high school students.

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