Explanation:
I assume the expression is
sqrt(x - y) + sqrt(x² - y²)
because what you actually wrote is
sqrt(x) - y + sqrt(x²) - y²
so, if my assumption is correct, please remember
(a + b)(a - b) = a² - b²
and with that we get
sqrt(x - y) + sqrt((x - y)(x + y)) =
= sqrt(x - y) + sqrt(x - y)×sqrt(x + y) =
= sqrt(x - y) × (1 + sqrt(x + y))
another possibility :
multiply everything by
(sqrt(x-y)-sqrt(x²-y²)) / (sqrt(x-y)-sqrt(x²-y²)) :
((x - y) - (x² - y²))/(sqrt(x-y)-sqrt(x²-y²))
or we square everything and then pull the square root of everything :
sqrt((x - y) + 2×sqrt(x-y)×sqrt(x²-y²) + (x² - y²)) =
= sqrt((x - y) + 2×sqrt(x-y)×sqrt((x-y)(x+y)) + (x² - y²)) =
= sqrt((x - y) + 2×sqrt(x-y)×sqrt(x-y)×sqrt(x+y) + (x² - y²)) =
= sqrt((x - y) + 2×(x-y)×sqrt(x+y) + (x² - y²)) =
these are the only things I can think of, if you truly wrote everything in the problem definition above.
but it is getting messier and messier.
frankly, the original expression is more simplified than any of the transformed result.
there is not much you can do with e.g. sqrt(x - y) except for squaring it.