Question

What is the square root of `48569834` ?

Answer 1

The square root is irrational.

Explanation:

`6969^2 = 48566961` and
`6970^2 = 48580900`.
Since 48,569,384 lies between two consecutive perfect squares, its square root is irrational.
If you need an approximation for it every computer comes with a calculator that will provide one. Alternatively, use Google.
Type sqrt(48569384), and Google's calculator will approximate the answer.

[Of course, every positive number has two square roots. The principle square root is positive, and the other square root is negative.]

Answer 2

`sqrt(48569384) = 6969+2423/(13938+2423/(13938+2423/(13938+...)))`

Explanation:

To attempt to find the square root of `48569384`, we can first split off pairs of digits starting from the right to get:

`48"|"56"|"93"|"84`

Note that `48 < 49 = 7^2`

Hence the square root is a little under `7000`

We can try using a method based on the Babylonian method to get a better approximation.

Starting with `p_0/q_0 = 7000/1`, to find better approximations for `sqrt(n)` where `n=48569384`, apply the following formulae:

`{ (p_(i+1) = p_i^2+nq_i^2), (q_(i+1) = 2p_i q_i) :}`

So:

`{ (p_1 = p_0^2+n q_0^2 = 7000^2+48569384*1^2 = 97569384), (q_1 = 2p_0 q_0 = 2*7000*1 = 14000) :}`

Let's see where this has got us after just one step:

`p_1/q_1 = 97569384/14000 ~~ 6969.2`

Try:

`6969^2 = 48566961 < 48569384`

`6970^2 = 48580900 > 48569384`

So the square root is somewhere strictly between `6969` and `6970`. It cannot be rational, so it's an irrational number.

Note that in general:

`sqrt(n) = a+b/(2a+b/(2a+b/(2a+...)))`

where `0 < a < sqrt(n)` and `b = n-a^2`

Now:

`48569384-6969^2 = 2423`

and:

`2*6969 = 13938`

So:

`sqrt(48569384) = 6969+2423/(13938+2423/(13938+2423/(13938+...)))`