Question

when a = ,√ 5+√3 ,& b=√5-√3, a/ b +b/a is equal to integer?​

Answer 1

We know that

a = √5+√3 and b = √5-√3

Now,

1/a = 1/(√5+√3) = (√5-√3)/2       Therefore, b/a = $(\sqrt{5} - \sqrt{3}) ^{2}$/2

1/b = 1/(√5-√3) = (√5+√3)/2       Therefore, a/b = $(\sqrt{5} + \sqrt{3}) ^{2}$/2

So, a/b + b/a = $(\sqrt{5} + \sqrt{3}) ^{2}$/2 + $(\sqrt{5} - \sqrt{3}) ^{2}$/2

                    = [$(\sqrt{5} + \sqrt{3}) ^{2}$ + $(\sqrt{5} - \sqrt{3}) ^{2}$]/2

                    = 16/2 = 8

For deriving last step use the formula

       $(a +b)^{2} + (a - b)^{2} = 2 (a^{2} + b^{2})$

Since answer is 8, it is an integer.