Question

How do you find the square root of 1849?

Answer 1

`sqrt(1849) = 43`

Explanation:

We could first seek to find the prime factorisation of `1849`, but as we shall see it is actually the square of a prime number, so that would be somewhat tedious.

Alternatively, let's split it into pairs of digits from the right to get:

`18"|"49`

Examining the leading `18`, note that it lies between `4^2` and `5^2`:

`4^2 = 16 < 18 < 25 = 5^2`

So:

`4 < sqrt(18) < 5`

and hence:

`40 < sqrt(1849) < 50`

To find a suitable correction, we can linearly interpolate between `40` and `50` to find:

`sqrt(1849) ~~ 40 + (50-40) * (1849 - 40^2)/(50^2-40^2)`

`color(white)(sqrt(1849)) ~~ 40 + 10 * (1849 - 1600)/(2500-1600)`

`color(white)(sqrt(1849)) ~~ 40 + 2490/900`

`color(white)(sqrt(1849)) ~~ 40 + 2.49 + 0.249 + 0.0249 +...`

`color(white)(sqrt(1849)) ~~ 42.76`

Hmmm... That's close to `43`, let's try `43^2`...

`43*43 = 40^2 + 2 * 40 * 3 + 3^2 = 1600+240+9 = 1849`

So:

`sqrt(1849) = 43`