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Number Theory: Interesting Results

Simplify -...

Question

Simplify $(- 3 + \sqrt{5}) (- 3 - \sqrt{5})$

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Solution

$(- 3 + \sqrt{)} (- 3 - \sqrt{5}) \Rightarrow - (\sqrt{5} - 3) (\sqrt{5} + 3) \Rightarrow - (5 - 9) \Rightarrow - 4$

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\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}

\frac{3 + \sqrt{5}}{3 - \sqrt{5}} + \frac{3 - \sqrt{5}}{3 + \sqrt{5}}

\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} + \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}

Q. Simplify the following:

\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}

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