1. Find the 2nd estimate for $\sqrt{341}$ using Newton's Iteration Method with initial estimate 18. Newton's formula: $$x_{n+1} = \frac{x_n + \frac{S}{x_n}}{2}$$ where $S$ is the number to find the root of. Calculate 2nd estimate: $$x_2 = \frac{18 + \frac{341}{18}}{2} = \frac{18 + 18.9444}{2} = \frac{36.9444}{2} = 18.4722$$ 2. Find the 3rd estimate for $\sqrt{341}$ using the 2nd estimate. Calculate 3rd estimate: $$x_3 = \frac{18.4722 + \frac{341}{18.4722}}{2} = \frac{18.4722 + 18.4603}{2} = \frac{36.9325}{2} = 18.4663$$ 3. Find the 2nd estimate for $\sqrt{558}$ using initial estimate 23. Calculate 2nd estimate: $$x_2 = \frac{23 + \frac{558}{23}}{2} = \frac{23 + 24.2609}{2} = \frac{47.2609}{2} = 23.6304$$ 4. Find the 3rd estimate for $\sqrt{558}$ using the 2nd estimate. Calculate 3rd estimate: $$x_3 = \frac{23.6304 + \frac{558}{23.6304}}{2} = \frac{23.6304 + 23.6093}{2} = \frac{47.2397}{2} = 23.6199$$ Final answers rounded to nearest thousandth: - $\sqrt{341} \approx 18.466$ - $\sqrt{558} \approx 23.620$