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

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UpStudy AI · Accepted Answer

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
$$
343.
$$
The simplified result is $$343$$.

Simplify, without using a calculator

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