Use linear approximation to estimate $sqrt[3]9.3$ correct to three decimal places. Write $f(x)=x^{1/3}$ and take the base point $a=8$.

Step-by-step solution

1. Let $f(x)=x^{1/3}$; then $f^{\prime }(x)=\frac{1}{3x^{2/3}}$
2. Evaluate at $a=8$: $f^{\prime }(8)=\frac{1}{3\cdot 8^{2/3}}=\frac{1}{3\cdot 4}=\frac{1}{12}$
3. Compute $f(8)=8^{1/3}=2$
4. Apply the linear approximation formula: $f(9.3)approxf(8)+f^{\prime }(8),(9.3-8)$
5. Substitute the values: $2+\frac{1}{12}\times 1.3=2+0.10833\approx 2.10833$
6. Round to three decimal places: $\boxed{2.108}$

Concept tested: Linear approximation for cube root

Published 7 October 2026 · How this solution was prepared and checked