Algebra

Simplify and rationalize the denominator.

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$\frac{1}{1+\frac{1}{\sqrt{3}+2}}\\ =\frac{1}{\frac{\sqrt{3}+2+1}{\sqrt{3}+2}}\\ =\frac{\sqrt{3}+2}{\sqrt{3}+3}\ |\ \times\frac{\sqrt{3}-3}{\sqrt{3}-3}\\ =\frac{3-3\cdot \sqrt{3} +2\cdot\sqrt{3} -6}{3-9}\\ \color{blue}=\frac{3+\sqrt{3}}{6}$

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$\frac{1}{1+\frac{1}{\sqrt{3}+2}}$

$\frac{1}{\frac{\sqrt{3}+3}{\sqrt{3}+2}}$     

$\frac{1}{\frac{b}{c}}=\frac{c}{b}$  $\frac{\sqrt{3}+2}{\sqrt{3}+3}$

$\frac{3+\sqrt{3}}{6}$