How do you find the square root of 85?

1 Answer
Mar 26, 2015

#sqrt85# is a non-terminating, non-repeating decimal. You can't finish finding it.
But you can get better and better approximations by: start with a number whose square is close to 85. (9 or 10. #9^2# is closer, so start there.

Now divide 85 by #9# to get #9.4444# (to 4 places)
Average the two numbers #9# and #9.4444#
#(9+9.4444)/2=9.2222#

Repeat:
#85-:9.2222=9.2169# (rounded to 4 places)
Average: #(9.2222+9.2169)/2=9.2196#

Repeat:
#85-:9.2196=9.2195# (rounded to 4 places)
Average: #(9.2196+9.2195)/2=9.2196#

#sqrt85~~9.2196#

That answer is accurate to 4 decimal places. If you need more accuracy, keep more decimal places at each step.