# How do you find the square root of 1242?

##### 3 Answers

It's

#### Explanation:

As we can see, 1242 is not a perfect square, and so therefore, will not be able to be simplified into a whole number. What we can do, though, is simplify

Let's start by dividing

That last expression can be rewritten as:

And that can be rewritten as:

Further simplification:

#### Explanation:

To find rational *approximations* to

First split

#12"|"42#

Examining the leftmost group of digits, note that:

#3^2 = 9 < 12 < 16 = 4^2#

So:

#3 < sqrt(12) < 4#

and:

#30 < sqrt(1242) < 40#

In fact note that

Let's use a variant on the Babylonian method:

Given a rational approximation

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

Starting with

Let

Then:

#{ (p_1 = p_0^2+n q_0^2 = 35^2+1242 * 1^2 = 1225+1242 = 2467), (q_1 = 2 p_1 q_1 = 2 * 35 * 1 = 70) :}#

#{ (p_2 = p_1^2+n q_1^2 = 2467^2 + 1242 * 70^2 = 6086089 + 6085800 = 12171889), (q_2 = 2 p_1 q_1 = 2 * 2467 * 70 = 345380) :}#

So:

#sqrt(1242) ~~ 12171889/345380 ~~ 35.2420204#

35.2420...

#### Explanation: