1. **Problem:** Given $a = -7i$ and $b = \frac{1}{6}i$, find $8(a + 2b)$. 2. **Formula and rules:** To solve this, use the distributive property and combine like terms. Remember that $i$ is the imaginary unit. 3. **Step-by-step solution:** - Calculate $2b$: $$2b = 2 \times \frac{1}{6}i = \frac{2}{6}i = \frac{1}{3}i$$ - Add $a$ and $2b$: $$a + 2b = -7i + \frac{1}{3}i = \left(-7 + \frac{1}{3}\right)i = \left(-\frac{21}{3} + \frac{1}{3}\right)i = -\frac{20}{3}i$$ - Multiply by 8: $$8(a + 2b) = 8 \times -\frac{20}{3}i = -\frac{160}{3}i$$ 4. **Final answer:** $$8(a + 2b) = -\frac{160}{3}i$$