Answer
$\approx 9.434$
Work Step by Step
We start with an approximation:
$a_0=9$
Compute $a_1,a_2,a_3,a_4,a_5$ using the formula:
$a_n=\dfrac{1}{2}\left(a_{n-1}+\dfrac{p}{a_{n-1}}\right)$, where $p=89$.
$a_1=\dfrac{1}{2}\left(a_0+\dfrac{89}{a_0}\right)=\dfrac{1}{2}\left(4+\dfrac{89}{4}\right)\approx 9.4444444$
$a_2=\dfrac{1}{2}\left(a_1+\dfrac{89}{a_1}\right)=\dfrac{1}{2}\left(9.4444444+\dfrac{89}{9.4444444}\right)\approx 9.4339869$
$a_3=\dfrac{1}{2}\left(a_2+\dfrac{89}{a_2}\right)=\dfrac{1}{2}\left(9.4339869+\dfrac{89}{9.4339869}\right)\approx 9.4339811$
$a_4=\dfrac{1}{2}\left(a_3+\dfrac{89}{a_3}\right)=\dfrac{1}{2}\left(9.4339811+\dfrac{89}{9.4339811}\right)\approx 9.4339811$
$a_5=\dfrac{1}{2}\left(a_4+\dfrac{89}{a_4}\right)=\dfrac{1}{2}\left(9.4339811+\dfrac{89}{9.4339811}\right)\approx 9.4339811$
So $\sqrt {89}\approx 9.434$
