Answer
See below.
Work Step by Step
Using a calculator:
$a_0=2$
$a_1=\frac{1}{2}(a_0+\frac{8}{a_0})=3$
$a_2=\frac{1}{2}(a_1+\frac{8}{a_1})\approx2.8333$
$a_3=\frac{1}{2}(a_2+\frac{8}{a_2})\approx2.82843$
$a_4=\frac{1}{2}(a_3+\frac{8}{a_3})\approx2.82842$
$a_5=\frac{1}{2}(a_4+\frac{8}{a_4})\approx2.82842$
$\sqrt{8}=2.82842712475$, so we got pretty close.