(36 1/2 + 16 1/4) ^ -2/3

First, calculate $36^{\frac{1}{2}}$.

$$36^{\frac{1}{2}} = \sqrt{36} = 6$$

Next, calculate $16^{\frac{1}{4}}$.

$$16^{\frac{1}{4}} = \sqrt[4]{16} = 2$$

$$(8)^{-\frac{2}{3}}$$

Use the property $a^{-m} = \frac{1}{a^m}$ to rewrite the expression.

$$(8)^{-\frac{2}{3}} = \frac{1}{8^{\frac{2}{3}}}$$

Use the property $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$.

$$8^{\frac{2}{3}} = (\sqrt[3]{8})^2$$

Calculate the cube root of 8.

$$\sqrt[3]{8} = 2$$

Square the result.

$$(2)^2 = 4$$

$$\frac{1}{8^{\frac{2}{3}}} = \frac{1}{4}$$

The final answer is $\frac{1}{4}$.