Question

Find the rational part of following expression:
$\frac{5+\sqrt{3}}{5-\sqrt{3}}$

• A. $\mathrm{1.272727...}$
• B. $\frac{14}{11}$
• C. $1\frac{6}{22}$
• D. All of the above.

Answer 1

The correct option is D All of the above.

On rationalising the denominator of $\frac{5+\sqrt{3}}{5-\sqrt{3}}$ we get,
$\Rightarrow \frac{5+\sqrt{3}}{5-\sqrt{3}}=\frac{5+\sqrt{3}}{5-\sqrt{3}}\times \frac{5+\sqrt{3}}{5+\sqrt{3}}=\frac{(5+\sqrt{3}{)}^2}{25-3}=\frac{25+3+10\sqrt{3}}{22}=\frac{28+10\sqrt{3}}{22}$

On separating the terms we get,
$\Rightarrow \frac{5+\sqrt{3}}{5-\sqrt{3}}=\frac{28}{22}+\frac{10\sqrt{3}}{22}$

So the rational part will be $\frac{28}{22}=\frac{14}{11}=\mathrm{1.272727......}$
Also,
$\frac{28}{22}=1\frac{6}{22}$
All the above are correct.