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!