Okay so for part one.... here's what I have now...

sqrt(27x^3y^7) + sqrt(48x^3y^7) This would be clearer if you wrote sqrt(27(x^3)(y^7)) + sqrt(48(x^3)(y^7)).

(sqrt(27)sqrt(x^3)sqrt(y^7)) + (sqrt(48)(x^3)(y^7)) I presume you mean (sqrt(27)sqrt(x^3)sqrt(y^7)) + (sqrt(48)sqrt(x^3)sqrt(y^7))

(3sqrt(3))(xsqrt(x))(y^3 sqrt(x)) + (2sqrt(12))(xsqrt(x))(y^3sqrt(y)) Yes although 2sqrt(12) = 4sqrt(3).

(3xy^3sqrt(3xy)) + (2xy^3sqrt(12xy)) Yes, but it would be clearer if you wrote (3x(y^3)sqrt(3xy)) + (2x(y^3)sqrt(12xy))

(5xy^3)sqrt(15xy) Here is where you go off the rails. Your next step should be

x(y^3)(3sqrt(3xy) + 2sqrt(12xy)) because x and y^3 are common factors. It is generally false that (a * sqrt(b)) + (c * sqrt(d)) = (a + c)(sqrt(b + d)). Try it with a couple of numbers (2 * sqrt(16)) + (7 * sqrt(9)) = (2 * 4) + (7 * 3) = 8 + 21 = 29, but (2 + 7) * sqrt(16 + 9) = 9 * sqrt(25) = 9 * 5 = 45.

Am I closer?