Nine square root root of seven minus square root root of seven minus square root root of seven

The given expression is:

\[
9\sqrt{\sqrt{7} - \sqrt{\sqrt{7} - \sqrt{\sqrt{7}}}}
\]

This expression involves nested square roots. Let's simplify it step by step.

### Step 1: Work from the innermost square root

Start by focusing on the innermost square root:

\[
\sqrt{7} - \sqrt{\sqrt{7}}
\]

Since there is no further simplification possible here (without knowing an exact value for \(\sqrt{7}\)), we can leave it as is. Now, consider the entire expression again:

\[
9 \times \sqrt{\sqrt{7} - \sqrt{\sqrt{7} - \sqrt{\sqrt{7}}}}
\]

### Step 2: Consider numerical approximation (optional)

To get an approximate value, we first compute the square roots numerically:

\[
\sqrt{7} \approx 2.645751
\]

Now we can approximate the nested square roots one step at a time:

1. \(\sqrt{\sqrt{7}} \approx \sqrt{2.645751} \approx 1.625\)
2. Substituting into the next layer: \(\sqrt{7} - \sqrt{1.625} \approx 2.645751 - 1.274 \approx 1.371\)
3. Finally, calculate \(\sqrt{1.371} \approx 1.17\)

Multiplying by 9:

\[
9 \times 1.17 \approx 10.53
\]

### Conclusion:
The approximate value of the given expression is \( \approx 10.53 \).

Would you like a more detailed breakdown or to verify any steps?

Here are 5 questions to consider:

1. How do we calculate the value of nested square roots?
2. Why are square roots of non-perfect squares irrational?
3. Can square roots be simplified when nested like this?
4. What are some real-world applications of square roots?
5. How does approximating square roots affect the precision of answers?

**Tip:** When dealing with nested square roots, start simplifying from the innermost root and work your way out!

Explore how to simplify the expression involving nine times the square root of 7 minus nested square roots. This detailed solution walks through the process step-by-step, using numerical approximation methods for square roots. Ideal for students in Grades 10-12.

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