Solve the following quadratic equations and simplify the surds and indices expression.

$$
x_1 = \frac{7 + 5}{6} = \frac{12}{6} = 2
$$
$$
x_2 = \frac{7 - 5}{6} = \frac{2}{6} = \frac{1}{3}
$$
The solutions are $\boxed{x = 2, x = \frac{1}{3}}$.

$$
x^2 + 6x = -5
$$

$$
x^2 + 6x + 9 = -5 + 9
$$
$$
(x + 3)^2 = 4
$$

$$
x + 3 = \pm\sqrt{4}
$$
$$
x + 3 = \pm 2
$$

$$
x = -3 \pm 2
$$
$$
x_1 = -3 + 2 = -1
$$
$$
x_2 = -3 - 2 = -5
$$
The solutions are $\boxed{x = -1, x = -5}$.

$$
\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}
$$
$$
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
$$

$$
5\sqrt{2} + 3(2\sqrt{2})
$$
$$
5\sqrt{2} + 6\sqrt{2}
$$
$$
11\sqrt{2}
$$
The simplified expression is $\boxed{11\sqrt{2}}$.

$$
2^{3 + (-5)}
$$
$$
2^{3 - 5}
$$
$$
2^{-2}
$$

$$
\frac{1}{2^2}
$$
$$
\frac{1}{4}
$$
The simplified expression is $\boxed{\frac{1}{4}}$.

$$
\log_{10}(100) = 2 \quad (\text{since } 10^2 = 100)
$$
$$
\log_{10}(0.001) = -3 \quad (\text{since } 10^{-3} = 0.001)
$$

$$
2 + (-3) = -1
$$
The simplified expression is $\boxed{-1}$.

$$
x = 2^5
$$

$$
x = 32
$$
The solution is $\boxed{x = 32}$.

$$
\sin^{-1}\left(\frac{1}{2}\right) = 30^\circ
$$