1. Use a calculator to approximate the square root of 320. Round to three decimal places

Answer 1

Calculators these days give square roots to 8-10 digits directly.

So I assume the question allows a calculator which does only the 4 operations, not including square root. If that is the case, this is how you can proceed:

We know that 18*18=324 (or check with the calculator).

So 18 is an approximation.

To improve the accuracy of an approximation x0 for √N, you calculate
x₁=x0-(N-xo²)/(2*x0)

for x0=18, N=320,
x1=18-(320-18²)/(2*18)
=17.889 (approx.)

Repeat the same calculation, using x1 instead of x0:
x2=17.889-(320-17.889²)/(2*17.889)
=17.888544

Clearly 17.889 was already accurate to 3 figures after the decimal.

Answer 2

Sorry, the above formula has a wrong sign, use:

x1=x0+(N-x0²)/(2*x0)
or better still
x1=x0-(x0²-N)/(2*x0)