Question 258748
let a be the larger of the numbers.
let b be the smaller of the numbers.


you have:


a-b = 4*sqrt(3)


you also have:


a*b = 8


the question is:


 What is the absolute value of the difference of their reciprocals?


their reciprocals are:


(1/a) for a and (1/b) for b


since a > b, then 1/b > 1/a


this is derived as follows:


a > b


divide both sides of the equation by ab to get:


1/b > 1/a


an example would be 3 > 2.


divide both sides of this equation by 6 to get:


1/2 > 1/3


the absolute value of the difference of their reciprocals is:


|1/b - 1/a| 


looking at the expression within the absolute value signs, we see:


(1/b - 1/a)


if we multiply this by (ab)/(ab), we get:


((a-b)/ab)


|1/b - 1/a| is therefore equal to |(a-b)/(ab)|


we know that a-b = 4*sqrt(3).


we also know that ab = 8


|1/b - 1/a| is therefore equal to |4*sqrt(3)/8|


this simplifies to |sqrt(3)/2|


since sqrt(3)/2 is always positive, then we have:


|1/b - 1/a| is equal to |sqrt(3)/2|


since sqrt(3)/2 is always positive, we have:


|1/b - 1/a| = |sqrt(3)/2| = sqrt(3)/2


this is because, by definition:


|x| = x if x is positive, and |x| = (-x) if x is negative.


your answer is:


the absolute value of the difference of their reciprocals equals sqrt(3)/2.