The first four digits of an eight-digit perfect square are 2012. Find its square root?

Apr 28, 2017

$\pm 2 \sqrt{503}$

Explanation:

$2012 = {2}^{2} \cdot 503$
And 503 is a prime number
Because ${22}^{2} < 503 < {23}^{2}$
So the square root of 2012 is
$\pm \sqrt{2012} = \pm 2 \sqrt{503}$

Apr 28, 2017

See below.

Explanation:

We have

$\sqrt{20120000} \approx 4485.53$

This square root can be extracted manually

https://en.wikipedia.org/wiki/Methods_of_computing_square_roots

and

${4485}^{2} = 20115225$

so the number is

$4486 \to {4486}^{2} = 20124196$