Evaluate the expression inside the square root.

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Step 1: 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: $\frac{3}{12} = \frac{1}{4}$ Step 2: Take the square root of the simplified fraction. $\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}}$ $= \frac{1}{2}$ The evaluated expression is $\boxed{\frac{1}{2}}$.

Step 1: Calculate the new hourly rate. 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. $New rate = Ksh. 210 \times \frac{8}{7}$ $New rate = Ksh. (210 \div 7) \times 8$ $New rate = Ksh. 30 \times 8$ $New rate = Ksh. 240 per hour$ Step 2: Convert the total working hours to an improper fraction. $10\frac{1}{2} hours = \frac{2 \times 10 + 1}{2} hours = \frac{21}{2} hours$ Step 3: Calculate the total earnings. $Total earnings = New rate \times Total hours$ $Total earnings = Ksh. 240 \times \frac{21}{2}$ $Total earnings = Ksh. (240 \div 2) \times 21$ $Total earnings = Ksh. 120 \times 21$ $Total earnings = Ksh. 2520$ Baraka would earn $\boxed{Ksh. 2520}$.

Step 1: Express all bases as powers of 2. 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: $(2^2)^{3x} \times 2^3 = (2^{-5})^{2x-1}$ Step 2: Apply the power rule $(a^m)^n = a^{mn}$. $2^{2 \times 3x} \times 2^3 = 2^{-5 \times (2x-1)}$ $2^{6x} \times 2^3 = 2^{-10x + 5}$ Step 3: Apply the product rule $a^m \times a^n = a^{m+n}$ on the left side. $2^{6x+3} = 2^{-10x + 5}$ Step 4: Since the bases are equal, equate the exponents. $6x + 3 = -10x + 5$ Step 5: Solve the linear equation for $x$. 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 $\boxed{x = \frac{1}{8}}$.

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