1. **State the problem:** Solve the exponential equation $$32^{2x - 1} = \frac{1}{8}$$. 2. **Rewrite bases as powers of 2:** Since 32 and 8 are powers of 2, write them as: $$32 = 2^5$$ $$8 = 2^3$$ 3. **Rewrite the equation using powers of 2:** $$\left(2^5\right)^{2x - 1} = \frac{1}{2^3}$$ 4. **Simplify the left side using power of a power rule:** $$2^{5(2x - 1)} = 2^{-3}$$ 5. **Set exponents equal since bases are the same:** $$5(2x - 1) = -3$$ 6. **Distribute 5:** $$10x - 5 = -3$$ 7. **Add 5 to both sides:** $$10x - \cancel{5} + 5 = -3 + 5$$ $$10x = 2$$ 8. **Divide both sides by 10:** $$\frac{\cancel{10}x}{\cancel{10}} = \frac{2}{10}$$ $$x = \frac{1}{5}$$ **Final answer:** $$x = \frac{1}{5}$$