Estimate the square root of 450 to the nearest hundreths place

Assume all in the head, no calculators.

450 is close to 21^2=441, so our initial estimate x0=21
a better estimate, using Newton's method,
x1=x0-(x0^2-450)/(2*x0)
=21-(21^2-450)/(2*21)
=21+9/42=21+3/14=21+1.5/7  [1.5/7 is easier in the head is easier to do than 9/42]
=21+0.21428
=21.21  (to two places)

Exact value is 21.2132034...