To solve this expression, we need to follow the order of operations (PEMDAS/BODMAS).

First, let's simplify the expression inside each square root:

√3 + √5 / √3 - √5 + √5 - √3 / √5 + √3

Next, let's simplify the division:

√3 + (√5 / √3) - √5 + (√5 - √3) / (√5 + √3)

Now, let's simplify the square roots:

√3 + (√(5/3)) - √5 + (√5 - √3) / (√5 + √3)

Next, let's simplify the subtraction:

√3 + (√(5/3)) - √5 + (√(5 - 3)) / (√5 + √3)

√3 + (√(5/3)) - √5 + (√2) / (√5 + √3)

Now, let's simplify the addition:

√3 + (√(5/3)) - √5 + (√2) / (√(5 + 3))

√3 + (√(5/3)) - √5 + (√2) / (√8)

Now, let's simplify the square root of 8:

√3 + (√(5/3)) - √5 + (√2) / (√(4 * 2))

√3 + (√(5/3)) - √5 + (√2) / (2√2)

Now, let's simplify the square root of 2:

√3 + (√(5/3)) - √5 + (√2) / (2 * √(2 * 1))

√3 + (√(5/3)) - √5 + (√2) / (2 * √2)

Now, let's simplify the multiplication:

√3 + (√(5/3)) - √5 + (√2) / (√4)

√3 + (√(5/3)) - √5 + (√2) / 2

Finally, let's simplify the square root of 4:

√3 + (√(5/3)) - √5 + (√2) / 2

So the simplified expression is √3 + (√(5/3)) - √5 + (√2) / 2.