Question
Simplify:
$\frac{2+\sqrt{3}}{2-\sqrt{3}}+\frac{2-\sqrt{3}}{2+\sqrt{3}}+\frac{\sqrt{3}-1}{\sqrt{3}+1}$
$\frac{2+\sqrt{3}}{2-\sqrt{3}}+\frac{2-\sqrt{3}}{2+\sqrt{3}}+\frac{\sqrt{3}-1}{\sqrt{3}+1}$
✓
Answer
$\frac{2+\sqrt{3}}{2-\sqrt{3}}+\frac{2-\sqrt{3}}{2+\sqrt{3}}+\frac{\sqrt{3}-1}{\sqrt{3}+1}$
$=\frac{2+\sqrt{3}}{2-\sqrt{3}}\times\frac{2+\sqrt{3}}{2+\sqrt{3}}\times\frac{2+\sqrt{3}}{2-\sqrt{3}}+\frac{2-\sqrt{3}}{2-\sqrt{3}}+\frac{\sqrt{3}-1}{\sqrt{3}+1}\times\frac{\sqrt{3}-1}{\sqrt{3}-1}$
$=\frac{\big(2+\sqrt{3}\big)^2}{2^2-\big(\sqrt{3}\big)^2}+\frac{\big(2-\sqrt{3}\big)^2}{2^2-\big(\sqrt{3}\big)^2}+\frac{\big(\sqrt{3}-1\big)^2}{\big(\sqrt{3}\big)^2-1}$
$=\frac{2^2+2\times2\sqrt{3}+\big(\sqrt{3}\big)^2}{4-3}+\frac{2^2-2\times2\sqrt{3}+\big(\sqrt{3}\big)^2}{4-3}+\frac{\big(\sqrt{3}\big)^2-2\sqrt{3}+1}{3-1}$
$=\frac{4+4\sqrt{3}+3}{1}+\frac{4-4\sqrt{3}+3}{1}+\frac{3-2\sqrt{3}+1}{2}$
$=7+4\sqrt{3}+7-4\sqrt{3}+\frac{4-2\sqrt{3}}{2}$
$=14+2-\sqrt{3}$
$=16-\sqrt{3}$
$=\frac{2+\sqrt{3}}{2-\sqrt{3}}\times\frac{2+\sqrt{3}}{2+\sqrt{3}}\times\frac{2+\sqrt{3}}{2-\sqrt{3}}+\frac{2-\sqrt{3}}{2-\sqrt{3}}+\frac{\sqrt{3}-1}{\sqrt{3}+1}\times\frac{\sqrt{3}-1}{\sqrt{3}-1}$
$=\frac{\big(2+\sqrt{3}\big)^2}{2^2-\big(\sqrt{3}\big)^2}+\frac{\big(2-\sqrt{3}\big)^2}{2^2-\big(\sqrt{3}\big)^2}+\frac{\big(\sqrt{3}-1\big)^2}{\big(\sqrt{3}\big)^2-1}$
$=\frac{2^2+2\times2\sqrt{3}+\big(\sqrt{3}\big)^2}{4-3}+\frac{2^2-2\times2\sqrt{3}+\big(\sqrt{3}\big)^2}{4-3}+\frac{\big(\sqrt{3}\big)^2-2\sqrt{3}+1}{3-1}$
$=\frac{4+4\sqrt{3}+3}{1}+\frac{4-4\sqrt{3}+3}{1}+\frac{3-2\sqrt{3}+1}{2}$
$=7+4\sqrt{3}+7-4\sqrt{3}+\frac{4-2\sqrt{3}}{2}$
$=14+2-\sqrt{3}$
$=16-\sqrt{3}$
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