15. Express each of the following as simply as possible with a rational denominator: a) √2/(√2 + √6); b) (1 + √3)/(2 + √3); c) (1 + √10)/(√10 − 3); d) (3 − √2)/(4 + 3√2); e) (1 − √2)/(3 − √8); f) (√3 − 5)/(2√3 − 4); g) (√12 + 3)/(3 − √3); h) (3√7 − 2)/(2√7 − 5).

Multiply by the conjugate of each denominator; first simplify √8 = 2√2 and √12 = 2√3 where needed.

1. a) √2/(√2 + √6) = 1/(1 + √3) = (√3 − 1)/2.
2. b) (1 + √3)/(2 + √3) = (1 + √3)(2 − √3) = √3 − 1.
3. c) (1 + √10)/(√10 − 3) = (1 + √10)(√10 + 3) = 13 + 4√10.
4. d) (3 − √2)/(4 + 3√2) = (3 − √2)(4 − 3√2)/(16 − 18) = (13√2 − 18)/2.
5. e) (1 − √2)/(3 − 2√2) = (1 − √2)(3 + 2√2) = −1 − √2.
6. f) (√3 − 5)/(2√3 − 4) = (√3 − 5)(√3 + 2)/[2(3 − 4)] = (7 + 3√3)/2.
7. g) (2√3 + 3)/(3 − √3) = (2√3 + 3)(3 + √3)/6 = (5 + 3√3)/2.
8. h) (3√7 − 2)/(2√7 − 5) = (3√7 − 2)(2√7 + 5)/(28 − 25) = (32 + 11√7)/3.