1. **State the problem:** Solve the equation $$16(3x-5) = \frac{1}{64}$$ for real numbers. 2. **Recall the formula and rules:** We want to isolate $x$. Note that $16$ and $\frac{1}{64}$ can be expressed as powers of 2: $$16 = 2^4, \quad \frac{1}{64} = 2^{-6}.$$ This helps simplify the equation. 3. **Rewrite the equation using powers of 2:** $$2^4(3x-5) = 2^{-6}$$ 4. **Divide both sides by $2^4$ to isolate $(3x-5)$:** $$3x-5 = \frac{2^{-6}}{2^4} = 2^{-6-4} = 2^{-10}$$ 5. **Express $2^{-10}$ as a fraction:** $$2^{-10} = \frac{1}{2^{10}} = \frac{1}{1024}$$ 6. **Solve for $x$:** $$3x - 5 = \frac{1}{1024}$$ 7. **Add 5 to both sides:** $$3x = 5 + \frac{1}{1024} = \frac{5 \times 1024}{1024} + \frac{1}{1024} = \frac{5120}{1024} + \frac{1}{1024} = \frac{5121}{1024}$$ 8. **Divide both sides by 3:** $$x = \frac{\frac{5121}{1024}}{3} = \frac{5121}{1024 \times 3} = \frac{5121}{3072}$$ 9. **Simplify the fraction if possible:** Since 5121 and 3072 share no common factors other than 1, the fraction is in simplest form. **Final answer:** $$x = \frac{5121}{3072}$$