1. **State the problem:** Solve the equation $$16^{3x-5} = \frac{1}{64}$$ in the set of real numbers. 2. **Recall the bases as powers of 2:** - $$16 = 2^4$$ - $$64 = 2^6$$ 3. **Rewrite the equation using powers of 2:** $$16^{3x-5} = \left(2^4\right)^{3x-5} = 2^{4(3x-5)}$$ $$\frac{1}{64} = 64^{-1} = \left(2^6\right)^{-1} = 2^{-6}$$ 4. **Set the exponents equal since the bases are the same:** $$4(3x-5) = -6$$ 5. **Simplify and solve for $x$:** $$12x - 20 = -6$$ $$12x = -6 + 20$$ $$12x = 14$$ $$x = \frac{14}{12}$$ 6. **Simplify the fraction:** $$x = \frac{\cancel{14}}{\cancel{12}} = \frac{7}{6}$$ **Final answer:** $$x = \frac{7}{6}$$