# How do you find the square root of #6724# ?

##### 1 Answer

#### Explanation:

We can try factoring

#color(white)(000000)6724#

#color(white)(000000)"/"color(white)(00)"\"#

#color(white)(00000)2color(white)(000)3362#

#color(white)(000000000)"/"color(white)(00)"\"#

#color(white)(00000000)2color(white)(000)1681#

#color(white)(000000000000)"/"color(white)(00)"\"#

#color(white)(00000000000)41color(white)(00)41#

So:

#sqrt(6724) = sqrt(2^2*41^2) = 2*41=82#

That "worked", but I cheated slightly in knowing that

Let's try another approach...

Given

#67|24#

Looking at the most significant pair of digits, note that:

#67 > 64 = 8^2#

Hence the square root of

We can use the Babylonian method to find a better approximation:

Given a number

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

So in our case, with

#a_1 = (80^2+6724)/(2*80) = (6400+6724)/160 = 82.025#

Hmmm - that's suspiciously close to

#82^2 = (80+2)^2 = 80^2+2*80*2+2^2 = 6400+320+4 = 6724#