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Write the reciprocal of

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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