1. The problem is to find the cube root of 32, which is written as $\sqrt[3]{32}$. 2. The cube root of a number $x$ is a number $y$ such that $y^3 = x$. 3. We want to find $y$ such that $y^3 = 32$. 4. We know that $2^3 = 2 \times 2 \times 2 = 8$ and $3^3 = 3 \times 3 \times 3 = 27$, so the cube root of 32 is between 3 and 4. 5. Check $4^3 = 4 \times 4 \times 4 = 64$, which is greater than 32, so the cube root is less than 4. 6. Since 32 is $2^5$, we can write $\sqrt[3]{32} = \sqrt[3]{2^5} = 2^{\frac{5}{3}}$. 7. Simplify $2^{\frac{5}{3}} = 2^{1 + \frac{2}{3}} = 2 \times 2^{\frac{2}{3}}$. 8. $2^{\frac{2}{3}} = (2^{\frac{1}{3}})^2 = (\sqrt[3]{2})^2$. 9. So the exact value is $2 \times (\sqrt[3]{2})^2$. 10. Numerically, $\sqrt[3]{32} \approx 3.1748$. Final answer: $\sqrt[3]{32} = 2 \times (\sqrt[3]{2})^2 \approx 3.1748$