1. **State the problem:** Solve the equation $$3 (8)^{2t - 1} = 3^{3x}$$ for $t$ and $x$. 2. **Rewrite the bases:** Note that $8 = 2^3$, so rewrite the equation as $$3 (2^3)^{2t - 1} = 3^{3x}$$ 3. **Simplify the exponent:** Using the power of a power rule, $$(2^3)^{2t - 1} = 2^{3(2t - 1)} = 2^{6t - 3}$$ 4. **Rewrite the equation:** Now the equation is $$3 imes 2^{6t - 3} = 3^{3x}$$ 5. **Analyze the equation:** The left side is a product of 3 and a power of 2, the right side is a power of 3. Since the bases are different and the left side is not a pure power of 3, the equation can only hold if both sides equal 0 or 1, or if the powers are adjusted accordingly. However, since 3 and 2 are prime bases, the only way for equality is if the powers of 3 on both sides match and the power of 2 equals 1. 6. **Set the powers of 3 equal:** The left side has a factor 3 (which is $3^1$), so rewrite as $$3^1 imes 2^{6t - 3} = 3^{3x}$$ 7. **Divide both sides by $3^1$:** $$rac{3^1 imes 2^{6t - 3}}{3^1} = rac{3^{3x}}{3^1}$$ 8. **Cancel common factors:** $$2^{6t - 3} = 3^{3x - 1}$$ 9. **Since the left side is a power of 2 and the right side is a power of 3, the only way for equality is if both sides equal 1:** $$2^{6t - 3} = 1 \quad \Rightarrow \quad 6t - 3 = 0$$ $$3^{3x - 1} = 1 \quad \Rightarrow \quad 3x - 1 = 0$$ 10. **Solve for $t$ and $x$:** $$6t - 3 = 0 \Rightarrow 6t = 3 \Rightarrow t = \frac{3}{6} = \frac{1}{2}$$ $$3x - 1 = 0 \Rightarrow 3x = 1 \Rightarrow x = \frac{1}{3}$$ **Final answer:** $$t = \frac{1}{2}, \quad x = \frac{1}{3}$$