a) To simplify the expression √15-√5 / 1-√3, we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is 1+√3.

(√15-√5) / (1-√3) * (1+√3) / (1+√3)

Simplifying the numerator and denominator separately:

Numerator: (√15-√5)(1+√3) = √15 + √45 - √5 - √15 = √45 - √5

Denominator: (1-√3)(1+√3) = 1 - √9 = 1 - 3 = -2

Therefore, the simplified expression is (√45 - √5) / -2.

b) To simplify the expression 2√3-√6 / √8-2, we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is √8+2.

(2√3-√6) / (√8-2) * (√8+2) / (√8+2)

Simplifying the numerator and denominator separately:

Numerator: (2√3-√6)(√8+2) = 2√24 + 4√3 - √48 - √12 = 2√24 + 4√3 - 4√3 - 2√3 = 2√24 - 2√3

Denominator: (√8-2)(√8+2) = 8 - 4 = 4

Therefore, the simplified expression is (2√24 - 2√3) / 4, which can be further simplified to √6 - √3 / 2.

c) To simplify the expression a-√a / 1-√a, we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is 1+√a.

(a-√a) / (1-√a) * (1+√a) / (1+√a)

Simplifying the numerator and denominator separately:

Numerator: (a-√a)(1+√a) = a + a√a - √a - a = a√a - √a

Denominator: (1-√a)(1+√a) = 1 - √a^2 = 1 - a

Therefore, the simplified expression is (a√a - √a) / (1 - a).

d) To simplify the expression x-2√x / √x-2 (x>=0, x≠0), we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is √x+2.

(x-2√x) / (√x-2) * (√x+2) / (√x+2)

Simplifying the numerator and denominator separately:

Numerator: (x-2√x)(√x+2) = x√x + 2x - 2x - 4√x = x√x - 4√x

Denominator: (√x-2)(√x+2) = x - 4

Therefore, the simplified expression is (x√x - 4√x) / (x - 4).