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

We start with the expression \[ \left(\frac{\sqrt{7^{2011}}-\sqrt{7^{2009}}}{\sqrt{7^{2008}}}+\sqrt{7}\right)^{2}. \] 1. Express the square roots as exponents: \[ \sqrt{7^n} = 7^{\frac{n}{2}}. \] Thus, \[ \sqrt{7^{2011}}=7^{\frac{2011}{2}},\quad \sqrt{7^{2009}}=7^{\frac{2009}{2}},\quad \sqrt{7^{2008}}=7^{\frac{2008}{2}}=7^{1004}. \] 2. Substitute these into the expression: \[ \left(\frac{7^{\frac{2011}{2}}-7^{\frac{2009}{2}}}{7^{1004}}+\sqrt{7}\right)^{2}. \] 3. Factor the numerator by taking out the common factor \(7^{\frac{2009}{2}}\): \[ 7^{\frac{2011}{2}}-7^{\frac{2009}{2}} = 7^{\frac{2009}{2}}\left(7^{\frac{2}{2}}-1\right)=7^{\frac{2009}{2}}(7-1)=6\cdot 7^{\frac{2009}{2}}. \] 4. Rewrite the fraction: \[ \frac{6\cdot 7^{\frac{2009}{2}}}{7^{1004}}. \] Notice that \[ 7^{\frac{2009}{2}}=7^{1004.5}, \] so the fraction becomes: \[ 6\cdot \frac{7^{1004.5}}{7^{1004}}=6\cdot 7^{0.5}=6\sqrt{7}. \] 5. Substitute this back into the original expression: \[ \left(6\sqrt{7}+\sqrt{7}\right)^{2} = \left(7\sqrt{7}\right)^{2}. \] 6. Square \(7\sqrt{7}\): \[ (7\sqrt{7})^2 = 7^2\cdot (\sqrt{7})^2 = 49\cdot 7 = 343. \] The simplified result is \[ 343. \]