1. Stated problem: Simplify the expression $c^1 \times \sqrt[3]{c^3}$.\n\n2. Break down the expression: $c^1$ is simply $c$. The term $\sqrt[3]{c^3}$ means the cube root of $c^3$.\n\n3. Apply the property of exponents and roots: $\sqrt[3]{c^3} = c^{3 \times \frac{1}{3}} = c^1 = c$.\n\n4. Now the expression becomes: $c \times c = c^{1+1} = c^2$.\n\n5. Final answer: $c^2$.