https://file.aiquickdraw.com/mathbot/file/1726315175753u2pco93i.jpg

The expression in the image is as follows:

\[
\sqrt{70 + \sqrt{100 + \sqrt{423 + \sqrt{298 + \sqrt{676 + 2}}}}}}
\]

Let's break it down step by step:

1. Start with the innermost square root: 
   \[
   \sqrt{676 + 2} = \sqrt{678}
   \]

   Approximating:
   \[
   \sqrt{678} \approx 26.03
   \]

2. Now move to the next square root:
   \[
   \sqrt{298 + 26.03} = \sqrt{324.03}
   \]

   Approximating:
   \[
   \sqrt{324.03} \approx 18.00
   \]

3. Move to the next:
   \[
   \sqrt{423 + 18} = \sqrt{441}
   \]

   \[
   \sqrt{441} = 21
   \]

4. Next step:
   \[
   \sqrt{100 + 21} = \sqrt{121}
   \]

   \[
   \sqrt{121} = 11
   \]

5. Finally:
   \[
   \sqrt{70 + 11} = \sqrt{81}
   \]

   \[
   \sqrt{81} = 9
   \]

So the value of the entire expression is \( 9 \).

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

Here are five related questions:
1. What are the properties of square roots that allow nested operations like these?
2. How does one approximate square roots for non-perfect squares, such as \( \sqrt{678} \)?
3. What is the general method to simplify nested radicals?
4. Could you provide examples of other complex expressions involving square roots?
5. How do you handle square roots when the radicand involves variables?

**Tip:** When approximating square roots of non-perfect squares, use nearby perfect squares to make a close estimate. For instance, \( \sqrt{678} \approx 26 \), because \( 26^2 = 676 \).

Solve the nested radical expression involving square roots: sqrt(70 + sqrt(100 + sqrt(423 + sqrt(298 + sqrt(676 + 2)))))

Discover how to simplify a complex nested radical expression step-by-step, including techniques for square root approximation. This example explores the use of properties of square roots and is ideal for high school students in Grades 9-12.

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