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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\(\frac{1}{1+\frac{1}{\sqrt{3}+2}}\)

\(\frac{1}{\frac{\sqrt{3}+3}{\sqrt{3}+2}}\)     

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

\(\frac{3+\sqrt{3}}{6}\)