Simplify, without using a calculator \( \left(\frac{\sqrt{7^{2011}}-\sqrt{7^{2009}}}{\sqrt{7^{2008}}}+\sqrt{7}\right)^{2} \)

To simplify the expression \(\left(\frac{\sqrt{7^{2011}}-\sqrt{7^{2009}}}{\sqrt{7^{2008}}}+\sqrt{7}\right)^{2}\), we start by rewriting the square roots: \[ \sqrt{7^{2011}} = 7^{1005.5}, \quad \sqrt{7^{2009}} = 7^{1004.5}, \quad \sqrt{7^{2008}} = 7^{1004} \] Now, we can express the original expression in terms of powers of 7. We rewrite the fraction: \[ \frac{\sqrt{7^{2011}} - \sqrt{7^{2009}}}{\sqrt{7^{2008}}} = \frac{7^{1005.5} - 7^{1004.5}}{7^{1004}} = \frac{7^{1004}(7 - 1)}{7^{1004}} = 7 - 1 = 6 \] Next, we can substitute this back into the original expression: \[ \frac{\sqrt{7^{2011}} - \sqrt{7^{2009}}}{\sqrt{7^{2008}}} + \sqrt{7} = 6 + \sqrt{7} \] Now we take this result and square it: \[ \left(6 + \sqrt{7}\right)^{2} \] Using the binomial expansion \((a + b)^2 = a^2 + 2ab + b^2\), we calculate: \[ = 6^2 + 2 \cdot 6 \cdot \sqrt{7} + (\sqrt{7})^2 = 36 + 12\sqrt{7} + 7 = 43 + 12\sqrt{7} \] Thus, the final simplified expression is: \[ \boxed{43 + 12\sqrt{7}} \]