1. **State the problem:** Simplify the expression $\frac{1}{3}x \cdot 0.36x$. 2. **Recall the multiplication rule for coefficients and variables:** When multiplying terms, multiply the coefficients (numbers) and then multiply the variables by adding their exponents. 3. **Multiply the coefficients:** $\frac{1}{3} \times 0.36 = 0.12$. 4. **Multiply the variables:** $x \times x = x^{1+1} = x^2$. 5. **Combine the results:** The product is $0.12x^2$. **Final answer:** $$0.12x^2$$