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: