# How do you find the exact square root of 39?

##### 1 Answer

#### Explanation:

If you would like to prove that

How do you prove that square root 15 is irrational?

You can find a succession of rational *approximations* for

Typically you would start with an approximation

#a_(i+1) = (a_i^2 + n) / (2a_i)#

where

I prefer to split

#p_(i+1) = p_i^2 + n q_i^2#

#q_(i+1) = 2 p_i q_i#

If the resulting

So in our case, let

Then:

#p_1 = p_0^2 + n q_0^2 = 6^2 + 39*1^2 = 36+39 = 75#

#q_1 = 2 p_0 q_0 = 2 * 6 * 1 = 12#

Now both of these are divisible by

#p_(1a) = 25#

#q_(1a) = 4#

If we stopped here we would get

Next iteration:

#p_2 = p_(1a)^2 + n q_(1a)^2 = 25^2 + 39*4^2 = 625 + 624 = 1249#

#q_2 = 2 p_(1a) q_(1a) = 2 * 25 * 4 = 200#

If we stopped here we would get

Next iteration:

#p_3 = p_2^2 + n q_2^2 = 1249^2 + 39*200^2 = 1560001 + 1560000 = 3120001#

#q_3 = 2 * 1249 * 200 = 499600#

If we stop here, we get the approximation:

#sqrt(39) ~~ 3120001 / 499600 ~~ 6.2449979983987#

Actually:

#sqrt(39) ~~ 6.24499799839839820584#