(i) Multiply by √3 – √2 both numerator and denominator.

\(\frac{\sqrt3-\sqrt2}{\sqrt3+\sqrt2}\)

\(= {(\sqrt3-\sqrt2)(\sqrt3-\sqrt2)\over (\sqrt3+\sqrt2)(\sqrt3-\sqrt2)}\)

\(= {(\sqrt3-\sqrt2)^2\over3-2}\)

\(=\frac {3-2\sqrt3\sqrt2+2}{1}\)

\(= 5-2\sqrt6\)

(ii)  Multiply by 7 – 4√3 both numerator and denominator.

 \(\frac{5+2\sqrt3}{7+4\sqrt3}\)

\(= \frac{(5+2\sqrt3)(7-4\sqrt3)} {(7+4\sqrt3)(7-4\sqrt3)}\)

\(= \frac{(5+2\sqrt3)(7-4\sqrt3)} {49-48}\)

\(= 36-20\sqrt3 + 14\sqrt3 – 24\)

\(= 11-6\sqrt3\)

(iii) Multiply by 3+2√2 both numerator and denominator.

 \(\frac{1+\sqrt2}{3-2\sqrt2}\)

\(= \frac{(1+\sqrt2)(3+2\sqrt2)}{(3-2\sqrt2)(3+2\sqrt2)}\)

 \(= \frac{(1+\sqrt2)(3+2\sqrt2)}{9-8}\)

\(= 3 +2\sqrt2 + 3\sqrt2 + 4\)

\(= 7 + 5\sqrt2\)

(iv) Multiply by 3√5 + 2√6 both numerator and denominator.

 \(\frac{2\sqrt6-\sqrt5}{3\sqrt5-2\sqrt6}\)

\(= \frac{(2\sqrt6 – \sqrt5)(3\sqrt5+2\sqrt6)}{(3\sqrt5-2\sqrt6)(3\sqrt5+2\sqrt6)}\)

\(= \frac{(2\sqrt6 – \sqrt5)(3\sqrt5+2\sqrt6)}{45-24}\)

\(= \frac{(2\sqrt6 – \sqrt5)(3\sqrt5+2\sqrt6)}{21}\)

\(= \frac{6\sqrt{30}+24-15-2\sqrt{30}}{21}\)

\(= \frac{4\sqrt{30}+9}{21}\)

(v) Multiply by √48 – √18 both numerator and denominator.

 \(\frac{4\sqrt3+5\sqrt2}{\sqrt{48}+\sqrt{18}}\)

\(= \frac{(4\sqrt3+5\sqrt2)(\sqrt{48}-\sqrt{18})}{(\sqrt{48}+\sqrt{18})(\sqrt{48}-\sqrt{18})}\)

\(= \frac{(4\sqrt3+5\sqrt2)(\sqrt{48}-\sqrt{18})}{48-18}\)

\(= \frac{48-12\sqrt6+20\sqrt6-30}{30}\)

\(=\frac{18+8\sqrt6}{30}\)

\(=\frac{9+4\sqrt6}{15}\)

(vi) Multiply by 2√2 – 3√3 both numerator and denominator.

 \(\frac{2\sqrt3-\sqrt5}{2\sqrt2+3\sqrt3}\)

\(= \frac{(2\sqrt3-\sqrt5)(2\sqrt2-3\sqrt3)}{(2\sqrt2+3\sqrt3)(2\sqrt2-3\sqrt3)}\)

\(= \frac{(2\sqrt3-\sqrt5)(2\sqrt2-3\sqrt3)}{8-27}\)

\(=\frac{(4\sqrt6-2\sqrt10)-(18+3\sqrt15)}{-19}\)

\(= \frac{(18-4\sqrt6+2\sqrt10-3\sqrt15)}{19}\)