1. **State the problem:** Simplify the expression $$\frac{8}{12} \times \frac{x^3}{t^2} \times \frac{t^3}{x^2}$$. 2. **Write the expression clearly:** $$\frac{8}{12} \times \frac{x^3}{t^2} \times \frac{t^3}{x^2}$$ 3. **Simplify the constants:** $$\frac{8}{12} = \frac{2 \times 4}{3 \times 4} = \frac{2}{3}$$ 4. **Combine the variables:** $$\frac{x^3}{t^2} \times \frac{t^3}{x^2} = \frac{x^3 \times t^3}{t^2 \times x^2} = \frac{x^{3-2} \times t^{3-2}}{1} = x^{1} t^{1} = x t$$ 5. **Multiply the simplified constants and variables:** $$\frac{2}{3} \times x t = \frac{2}{3} x t$$ 6. **Final answer:** $$\boxed{\frac{2}{3} x t}$$