Question

Rationalise the denominator of $\frac{2+\sqrt{5}}{2-\sqrt{5}}$

• A. $-(9-2\sqrt{5})$
• B. $-(9+4\sqrt{5})$
• C. $9-4\sqrt{5}$
• D. $4-9\sqrt{5}$

Answer 1

The correct option is B

$-(9+4\sqrt{5})$

The least rationalising factor of $\frac{2+\sqrt{5}}{2-\sqrt{5}}$ is $2+\sqrt{5}$

Multiplying both the numerator and the denominator by $2+\sqrt{5}$, we get

$\frac{2+\sqrt{5}}{2-\sqrt{5}}\times \frac{2+\sqrt{5}}{2+\sqrt{5}}$

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

$=\frac{4+5+4\sqrt{5}}{4-5}$

$=\frac{9+4\sqrt{5}}{-1}$

$=-(9+4\sqrt{5})$