Without using a calculator, simplify. Clearly show your working. 11 sqrt(536) + 534130

Here's how to simplify the expression without a calculator:

Step 1: Factor out the common term in the numerator. The common factor in $5^{36} + 5^{34}$ is $5^{34}$. $5^{36} + 5^{34} = 5^{34} \cdot 5^2 + 5^{34} \cdot 1 = 5^{34}(5^2 + 1)$

Step 2: Simplify the term inside the parenthesis. $5^2 + 1 = 25 + 1 = 26$

Step 3: Substitute this back into the expression under the square root. The expression becomes: $11 \sqrt{\frac{5^{34}(26)}{130}}$

Step 4: Simplify the fraction inside the square root. Notice that $130 = 5 \times 26$. $\frac{26}{130} = \frac{26}{5 \times 26} = \frac{1}{5}$

Step 5: Rewrite the expression under the square root using the simplified fraction. $\frac{5^{34}}{5} = 5^{34-1} = 5^{33}$ So the expression is now: $11 \sqrt{5^{33}}$

Step 6: Simplify the square root of $5^{33}$. We can write $5^{33}$ as $5^{32} \cdot 5^1$. $\sqrt{5^{33}} = \sqrt{5^{32} \cdot 5} = \sqrt{5^{32}} \cdot \sqrt{5}$ Since $\sqrt{5^{32}} = (5^{32})^{1/2} = 5^{32/2} = 5^{16}$. So, $\sqrt{5^{33}} = 5^{16} \sqrt{5}$.

Step 7: Combine with the leading factor of 11. $11 \cdot 5^{16} \sqrt{5}$

The simplified expression is $\boxed{11 \cdot 5^{16} \sqrt{5}}$.

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