How do you rationalize the denominator and simplify 1/(1+sqrt(3)-sqrt(5))?

Mar 30, 2016

$\frac{7 + 3 \sqrt{3} + \sqrt{5} + 2 \sqrt{15}}{11}$ = 2.0162, nearly.

Explanation:

Multiply numerator and denominator by $\left(1 + \sqrt{3} + \sqrt{5}\right)$.
Use $\left(a + b\right) \left(a - b\right) = \left({a}^{2} - {b}^{2}\right)$ for the denominator.

The denominator now is ${\left(1 + \sqrt{3}\right)}^{2} - {\left(\sqrt{5}\right)}^{2} = 2 \sqrt{3} + 1$.

Multiply the numerator and denominator by the conjugate $2 \sqrt{3} + 1$ of the denominator.

Now, the denominator is rationalized to 11.

Expand the product in the numerator and simplify for the answer.

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