2011/093) If a=4 root 6 /(root 2+ root 3) then find value of (a+2 root 2)/(a-2 root 2) + (a+ 2 root 3)/(a - 2 root 3)?

this is a perfect example of using componendo dividendo method

a=4 root 6 /(root 2+ root 3)

so a/(2 root 2) = 2 root3/(root 2 + root 3)

by componendo dividendo

( a+ 2 root 2)/(a- 2 root 2) = (2 root 3 + root 2 + root 3)/(root 3 - root 2)

= (3 root 3 + root 2) /(root 3 - root2) ...1

again for the second term

a=4 root 6 /(root 2+ root 3)
so a/(2 root 3) = 2 root2/(root 2 + root 3)

by componendo dividendo

( a+ 2 root 3)/(a- 2 root 3) = (2 root 2 + root 2 + root 3)/(root 2 - root 3)
= ( 3 root 2 + root 3)/ (root 2 - root 3) = - (- root 3 - 3 root 2)/(root 3 - root 2) .. 2

add (1) and (2) to get
( a+ 2 root 2)/(a- 2 root 2) + ( a+ 2 root 3)/(a- 2 root 3)
= 2(root 3 – root 2)/(root 3 – root 2) = 2


(comonendo and dividendo: for the persons who are not familiar

if a/b = c/d then (a+b)/(a-b) = (c+d)/(c-d))

for a proof of it (a+b)/(a-b) = (a/b +1)/(a/b-1)= (c/d + 1)/(c/d - 1) = (c+d)/(c-d))