Question

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

Answer 1

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`.

Answer 2

`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`