1. **State the problem:** Find the smallest zero of the function $f(x) = x^5 (x - 1)^2 (x + 8)$ and its multiplicity. 2. **Identify zeros:** The zeros are the values of $x$ that make each factor zero. - From $x^5$, zero is $x=0$ with multiplicity 5. - From $(x - 1)^2$, zero is $x=1$ with multiplicity 2. - From $(x + 8)$, zero is $x=-8$ with multiplicity 1. 3. **Find the smallest zero:** Among $-8$, $0$, and $1$, the smallest is $-8$. 4. **Multiplicity of smallest zero:** The factor $(x + 8)$ has exponent 1, so multiplicity is 1. **Final answer:** The smallest zero is $-8$ with multiplicity 1.