Question:
Write the reciprocal of $(2+\sqrt{3})$.
Solution:
The reciprocal of $(2+\sqrt{3})$
$=\frac{1}{(2+\sqrt{3})}$
$=\frac{1}{(2+\sqrt{3})} \times \frac{(2-\sqrt{3})}{(2-\sqrt{3})}$
$=\frac{(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})}$
$=\frac{(2-\sqrt{3})}{2^{2}-(\sqrt{3})^{2}} \quad\left[(a+b)(a-b)=a^{2}-b^{2}\right]$
$=\frac{(2-\sqrt{3})}{4-3}$
$=\frac{(2-\sqrt{3})}{1}$
$=(2-\sqrt{3})$
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