1. The problem states that the area of a triangle is $\frac{32}{15}$ square centimeters and the height is $\frac{3}{4}$ centimeters. We need to find the length of the base. 2. Recall the formula for the area of a triangle: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. Substitute the known values into the formula: $$\frac{32}{15} = \frac{1}{2} \times \text{base} \times \frac{3}{4}$$ 4. Simplify the right side: $$\frac{32}{15} = \frac{1}{2} \times \frac{3}{4} \times \text{base} = \frac{3}{8} \times \text{base}$$ 5. To isolate the base, divide both sides by $\frac{3}{8}$: $$\text{base} = \frac{\frac{32}{15}}{\frac{3}{8}}$$ 6. Division of fractions means multiplying by the reciprocal: $$\text{base} = \frac{32}{15} \times \frac{8}{3}$$ 7. Multiply numerators and denominators: $$\text{base} = \frac{32 \times 8}{15 \times 3} = \frac{256}{45}$$ 8. Check if the fraction can be simplified. The numerator is $256 = 2^8$ and the denominator is $45 = 3^2 \times 5$. No common factors, so the fraction is in simplest form. 9. Therefore, the length of the base is: $$\boxed{\frac{256}{45}}$$