Question

How do you find the square root of 6889?

Answer 1

`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`