Answer
$\approx 9.43398115$
Work Step by Step
We apply the given recursive formula to find the approximate value for $\sqrt {89}$ as follows:
$$\displaystyle a_0=9 \\ a_1=\frac{1}{2}(a_0+\frac{89}{a_0})=9.44444 \\ a_2=\frac{1}{2}(a_1+\frac{89}{a_1})\approx9.433986 \\ a_3=\frac{1}{2}(a_2+\frac{89}{a_2})\approx9.433981 \\ a_4=\frac{1}{2}(a_3+\frac{89}{a_3})\approx9.4339811 \\a_5=\frac{1}{2}(a_4+\frac{89}{a_4})\approx 9.43398115$$
Therefore, we get the approximate value for $\sqrt{89}=9.43398113206 \approx 9.43398115$
