How do you find the square root of 6889?

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How do you find the square root of 6889?

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Answer 1 · 2016-06-28T22:29:24

$sqrt(6889) = 83$

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:

$sqrt(6889) = 83$

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How do you find the square root of 6889?

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