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

Rationalise each denominator by multiplying the numerator and denominator by the conjugate.

1. a) 1/(√2 + 1) = √2 − 1.
2. b) 4/(√3 − 1) = 4(√3 + 1)/2 = 2√3 + 2.
3. c) 1/(√6 − 2) = (√6 + 2)/2 = (√6 + 2)/2.
4. d) 3/(2 + √3) = 3(2 − √3) = 6 − 3√3.
5. e) 1/(2 + √5) = √5 − 2.
6. f) √2/(√2 − 1) = √2(√2 + 1) = 2 + √2.
7. g) 6/(√7 + 3) = 6(3 − √7)/2 = 9 − 3√7.
8. h) 1/(3 + 2√2) = 3 − 2√2.
9. i) 1/(4 − 2√3) = (4 + 2√3)/4 = 1 + √3/2.
10. j) 3/(3√2 + 4) = 3(3√2 − 4)/2 = (9√2 − 12)/2.
11. k) 2√3/(7 − 4√3) = 2√3(7 + 4√3) = 24 + 14√3.
12. l) 6/(√5 − √3) = 6(√5 + √3)/2 = 3√5 + 3√3.