How do you find the square root of 6889?

1 Answer
Jun 28, 2016

Answer:

#sqrt(6889) = 83#

Explanation:

Note that #10^2=100#, so if we repeatedly divide by #100# until we get a number less than #100#, then its square root multiplied by a power of #10# will be the square root fo the original number.

In our example, we only need to divide #6889# by #100# once to get a number less than #100#, viz #68.89#.

Hopefully we know the first #10# square numbers, so we can tell:

#8^2 = 64 < 68.89 < 81 = 9^2#

Hence:

#8 < sqrt(68.89) < 9#

and:

#80 < sqrt(6889) < 90#

We can linearly interpolate to get closer.

Linearly interpolating in this way is approximating part of the parabola of #x^2# with a straight line segment.

#sqrt(6889) ~~ 80 + (6889-80^2)/(90^2-80^2)*(90-80)#

#=80 + (6889-6400)/(8100-6400)*(90-80)#

#=80+4890/1700#

#~~82.88#

Hmmm. That's quite close to #83#. What is #83^2#?

#83^2 = 6889#

So:

#sqrt(6889) = 83#