Question

Simplify; (2/root5 + root3) + (1/root3 + root2)

Answer 1

Answer:

Step-by-step explanation:

i) \frac{2}{\sqrt{5}+\sqrt{3}}\\=\frac{2(\sqrt{5}-\sqrt{3})}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})}\\=\frac{2(\sqrt{5}-\sqrt{3})}{(\sqrt{5})^{2}-(\sqrt{3})^{2}}\\=</p><p>\frac{2(\sqrt{5}-\sqrt{3})}{(5-3)}\\=\frac{2(\sqrt{5}-\sqrt{3})}{2}\\=\sqrt{5}-\sqrt{3}---(1)

ii) \frac{1}{\sqrt{3}+\sqrt{2}}\\=\frac{(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}\\=\frac{(\sqrt{3}-\sqrt{2})}{(\sqrt{3})^{2}-(\sqrt{2})^{2}}\\=</p><p>\frac{(\sqrt{3}-\sqrt{2})}{(3-2)}\\=\frac{(\sqrt{3}-\sqrt{2})}{1}\\=\sqrt{3}-\sqrt{2}---(2)

iii) \frac{3}{\sqrt{5}+\sqrt{2}}\\=\frac{3(\sqrt{5}-\sqrt{2})}{(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})}\\=\frac{3(\sqrt{5}-\sqrt{2})}{(\sqrt{5})^{2}-(\sqrt{2})^{2}}\\=</p><p>\frac{3(\sqrt{5}-\sqrt{2})}{(5-2)}\\=\frac{3(\sqrt{5}-\sqrt{2})}{3}\\=\sqrt{5}-\sqrt{2}---(3)

Now,\\\frac{2}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{2}}-\frac{3}{\sqrt{5}+\sqrt{2}}\\=\sqrt{5}-\sqrt{3}+\sqrt{3}-\sqrt{2}-(\sqrt{5}-\sqrt{2})\\ [from \\(1),(2)\:and\:(3)]

=\sqrt{5}-\sqrt{3}+\sqrt{3}-\sqrt{2}-\sqrt{5}+\sqrt{2}\\=0