# How do you find the square root of 6889?

Jun 28, 2016

$\sqrt{6889} = 83$

#### 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.

$\sqrt{6889} \approx 80 + \frac{6889 - {80}^{2}}{{90}^{2} - {80}^{2}} \cdot \left(90 - 80\right)$

$= 80 + \frac{6889 - 6400}{8100 - 6400} \cdot \left(90 - 80\right)$

$= 80 + \frac{4890}{1700}$

$\approx 82.88$

Hmmm. That's quite close to $83$. What is ${83}^{2}$?

${83}^{2} = 6889$

So:

$\sqrt{6889} = 83$