Explain why the square root of 89 must be between 9 and 10?

2 Answers
Jun 1, 2016

Answer:

Evaluating the square of #9# and #10#.

Explanation:

Recall that the square root is the inverse operation of the square.
Then we can see that the square of #9# is

#9^2=81#

and the square of #10# is

#10^2=100#

comparing with #89# we have that #89>81# and #89<100#.
Doing the square root we have

#sqrt(89) > sqrt(81)# that is #sqrt(89)>9#

#sqrt(89) < sqrt(100)# that is #sqrt(89)<10#.
We proved that the square root of #89# is greater than #9# and smaller than #10#.

Jun 1, 2016

Answer:

#9 < sqrt(89) < 10#

Explanation:

We have #81 = 9^2 < 89 < 10^2 = 100#
and #sqrt(x)# is monotonic for # 0 < x < infty #
which means that if #0 < x_1 < x_2->sqrt(x_1) < sqrt(x_2)#
then #9 < sqrt(89) < 10#