What is the square root of 48569834 ?

2 Answers
Apr 4, 2017

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.]

Apr 4, 2017

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+...)))