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

The common factor in $5^{36} + 5^{34}$ is $5^{34}$.

$$5^{36} + 5^{34} = 5^{34} \cdot 5^2 + 5^{34} \cdot 1 = 5^{34}(5^2 + 1)$$

The expression becomes:

$$11 \sqrt{\frac{5^{34}(26)}{130}}$$

Notice that $130 = 5 \times 26$.

$$\frac{26}{130} = \frac{26}{5 \times 26} = \frac{1}{5}$$

$$\frac{5^{34}}{5} = 5^{34-1} = 5^{33}$$

So the expression is now:

$$11 \sqrt{5^{33}}$$

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, $\sqrt{5^{33}} = 5^{16} \sqrt{5}$.