Solution:
Problem 1.
[tex] \frac{3}{ \sqrt{5} - 2 } [/tex]
Rationalize the denominator by multiplying the numerator and denominator by the denominator. (Important: if the denominator has a minus sign, change to plus and vice versa)
[tex]⟹\frac{3 \times \sqrt{5 + 2} }{ (\sqrt{5} - 2 ) \times ( \sqrt{5} + 2) } [/tex]
[tex] ⟹ \frac{3 \sqrt{5} + 6 }{ \sqrt{25} + 2 \sqrt{5} - 2 \sqrt{5} - 4 } ⟹\frac{3 \sqrt{5} + 6 }{5 - 4} [/tex]
[tex] ⟹\frac{3 \sqrt{5} + 6}{1} ⟹3 \sqrt{5} + 6[/tex]
Problem 2.
[tex] \frac{6 - \sqrt{12} }{ \sqrt{3} + \sqrt{10} } [/tex]
Rationalize the denominator.
[tex] ⟹\frac{(6 - \sqrt{12} ) \times ( \sqrt{3} - \sqrt{10} )}{ (\sqrt{3} + \sqrt{10} ) \times ( \sqrt{3} - \sqrt{10} } [/tex]
[tex]⟹\frac{6 \sqrt{3} - 6 \sqrt{10} - \sqrt{36} + \sqrt{120} }{ \sqrt{9} - \sqrt{30} + \sqrt{30} - \sqrt{100} } [/tex]
[tex]⟹ \frac{6 \sqrt{3} - 6 \sqrt{10} - \sqrt{36} + \sqrt{120} }{3 - 10} ⟹ \frac{6 \sqrt{3} - 6 \sqrt{10} - \sqrt{36} + \sqrt{120} }{ - 7} [/tex]
[tex] ⟹\frac{6 \sqrt{3} - 6 \sqrt{10} - 6 + \sqrt{12 \times 10} }{ - 7}⟹\frac{6 \sqrt{3} - 6 \sqrt{10} - 6 + 4 \sqrt{3 \times 10} }{ - 7} [/tex]
[tex] ⟹ - \frac{6 \sqrt{3} - 6 \sqrt{10} - 6 + 2 \sqrt{30} }{7} [/tex]
Therefore:
1. [tex]3 \sqrt{5} + 6[/tex]
2. [tex] - \frac{6 \sqrt{3} - 6 \sqrt{10} - 6 + 2 \sqrt{30} }{7} [/tex]
I hope these help
