Question

Write an equivalent expression by rationalizing the denominator $\frac{1}{\sqrt{5}+\sqrt{3}}$

• A. $\frac{\sqrt{5}-\sqrt{3}}{2}$
  
• B. $\frac{\sqrt{5}+\sqrt{3}}{2}$
  
• C. (a) and (b) above.
• D. None of the above

Answer 1

The correct option is A $\frac{\sqrt{5}-\sqrt{3}}{2}$


To rationalize the denominator of $\frac{1}{\sqrt{5}+\sqrt{3}}$, multiply the denominator with its conjugate pair.
The conjugate pair of $\sqrt{5}+\sqrt{3}$ is $\sqrt{5}-\sqrt{3}$.

Let's multiply both the numerator and denominator with $\sqrt{5}-\sqrt{3}$.

$\frac{1}{\sqrt{5}+\sqrt{3}}=\frac{1}{\sqrt{5}+\sqrt{3}}\times 1$
$=\frac{1}{\sqrt{5}+\sqrt{3}}\times \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}$

$=\frac{\sqrt{5}-\sqrt{3}}{(\sqrt{5}+\sqrt{3})\times (\sqrt{5}-\sqrt{3})}$

$=\frac{\sqrt{5}-\sqrt{3}}{(\sqrt{5}{)}^2-(\sqrt{3}{)}^2}$

$=\frac{\sqrt{5}-\sqrt{3}}{5-3}$ $=\frac{\sqrt{5}-\sqrt{3}}{2}$