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4.
1) $\frac{ x^{2}-2 }{x+ \sqrt{2} }$ = $\frac{(x+ \sqrt{2}) * (x- \sqrt{2}) }{x+ \sqrt{2} }$ = x- $\sqrt{2}$
2) $\frac{ \sqrt{2}+ 2}{ \sqrt{2} }$ = $\frac{ \sqrt{2} * (1+ \sqrt{2}) }{ \sqrt{2} }$ = 1+ $\sqrt{2}$
3) $\frac{ \alpha + \sqrt \alpha{}}{ \alpha \sqrt{ \alpha } + \alpha }$ = $\frac{ \sqrt{ \alpha } * ( \sqrt{ \alpha }+ 1) }{ \alpha * ( \sqrt{ \alpha } +1) }$ = $\frac{ \sqrt{ \alpha } }{ \alpha }$ = $\frac{1}{ \sqrt{ \alpha } }$
5.
1) $\frac{4}{ \sqrt{3} + 1 } = \frac{4 * ( \sqrt{3} + 1) }{ 3+1} = \frac{4 * (\sqrt{3} +1) }{4} = \sqrt{3} + 1$
2) $\frac{2}{ \sqrt{5} - \sqrt{3} } = \frac{2* ( \sqrt{5} - \sqrt{3)} }{ 5-3} = \frac{2* (\sqrt{5} - \sqrt{3} ) }{2} = \sqrt{5} - \sqrt{3}$
3) $\frac{4}{ \sqrt{ \alpha } + \sqrt{b} } = \frac{4 * ( \sqrt{a} + \sqrt{b}) }{ a +b} = \frac{4 \sqrt{a} + 4 \sqrt{b} }{ a + b}$