# What is the square root of #48569834# ?

##### 2 Answers

The square root is irrational.

#### Explanation:

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.]

#### Explanation:

To attempt to find the square root of

#48"|"56"|"93"|"84#

Note that

Hence the square root is a little under

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

Starting with

#{ (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

Note that in general:

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

where

Now:

#48569384-6969^2 = 2423#

and:

#2*6969 = 13938#

So:

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