1. **State the problem:** Solve for $x$ in the equation $$64 = 32^{x - 3}$$. 2. **Recall the formula and rules:** We want to express both sides with the same base to compare exponents. Both 64 and 32 are powers of 2: $$64 = 2^6$$ $$32 = 2^5$$ 3. **Rewrite the equation using base 2:** $$2^6 = (2^5)^{x - 3}$$ 4. **Apply the power of a power rule:** $$(2^5)^{x - 3} = 2^{5(x - 3)}$$ So the equation becomes: $$2^6 = 2^{5(x - 3)}$$ 5. **Since the bases are equal, set the exponents equal:** $$6 = 5(x - 3)$$ 6. **Solve for $x$:** $$6 = 5x - 15$$ Add 15 to both sides: $$6 + 15 = 5x$$ $$21 = 5x$$ Divide both sides by 5: $$\frac{21}{5} = x$$ Show cancellation: $$x = \cancel{\frac{21}{\cancel{5}}}$$ 7. **Final answer:** $$x = \frac{21}{5}$$