Evaluate the expression inside the square root.

To combine the fractions, find a common denominator, which is 12.

$$\frac{1}{12} - \frac{1}{3} + \frac{1}{2} = \frac{1}{12} - \frac{1 \times 4}{3 \times 4} + \frac{1 \times 6}{2 \times 6}$$ $$= \frac{1}{12} - \frac{4}{12} + \frac{6}{12}$$ $$= \frac{1 - 4 + 6}{12}$$ $$= \frac{3}{12}$$

Simplify the fraction:

$$\frac{3}{12} = \frac{1}{4}$$

$$\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}}$$ $$= \frac{1}{2}$$

The evaluated expression is $\boxed{\frac{1}{2}}$.

The original rate is Ksh. 210 per hour. The rate changed in the ratio 8:7, meaning the new rate is $\frac{8}{7}$ times the old rate.

$$\text{New rate} = \text{Ksh. } 210 \times \frac{8}{7}$$ $$\text{New rate} = \text{Ksh. } (210 \div 7) \times 8$$ $$\text{New rate} = \text{Ksh. } 30 \times 8$$ $$\text{New rate} = \text{Ksh. } 240 \text{ per hour}$$

$$\text{Total earnings} = \text{New rate} \times \text{Total hours}$$ $$\text{Total earnings} = \text{Ksh. } 240 \times \frac{21}{2}$$ $$\text{Total earnings} = \text{Ksh. } (240 \div 2) \times 21$$ $$\text{Total earnings} = \text{Ksh. } 120 \times 21$$ $$\text{Total earnings} = \text{Ksh. } 2520$$

Baraka would earn $\boxed{\text{Ksh. } 2520}$.

The given equation is $4^{3x} \times 8 = \left(\frac{1}{32}\right)^{2x-1}$.
We know that $4 = 2^2$, $8 = 2^3$, and $32 = 2^5$.
Therefore, $\frac{1}{32} = 2^{-5}$.
Substitute these into the equation:

$$(2^2)^{3x} \times 2^3 = (2^{-5})^{2x-1}$$

$$2^{2 \times 3x} \times 2^3 = 2^{-5 \times (2x-1)}$$ $$2^{6x} \times 2^3 = 2^{-10x + 5}$$

Add $10x$ to both sides:

$$6x + 10x + 3 = 5$$ $$16x + 3 = 5$$

Subtract 3 from both sides:

$$16x = 5 - 3$$ $$16x = 2$$

Divide by 16:

$$x = \frac{2}{16}$$ $$x = \frac{1}{8}$$

The solution is $\boxed{x = \frac{1}{8}}$.