Question

Simplify : $\frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}.$

• A.
  $\frac{40}{11}$
• B.
  $\frac{42}{11}$
• C.
  $\frac{43}{11}$
• D.
  $\frac{39}{11}$

Answer 1

The correct option is B

$\frac{42}{11}$
$\frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}=(\frac{4+\sqrt{5}}{4-\sqrt{5}}\times \frac{4+\sqrt{5}}{4+\sqrt{5}})+(\frac{4-\sqrt{5}}{4+\sqrt{5}}\times \frac{4-\sqrt{5}}{4-\sqrt{5}})=\frac{(4{)}^2+(\sqrt{4}{)}^2+2\times 4\times \sqrt{5}}{(\sqrt{4}{)}^2-(\sqrt{5}{)}^2}+\left(\frac{(4{)}^2+(\sqrt{5}{)}^2-2\times 4\times \sqrt{5}}{(\sqrt{4}{)}^2-(\sqrt{5}{)}^2}\right)=\frac{16+5+8\sqrt{5}}{16-5}+\frac{16+5-8\sqrt{5}}{16-5}=\frac{21+8\sqrt{5}}{11}+\frac{21-8\sqrt{5}}{11}=\frac{21+8\sqrt{5}+21-8\sqrt{5}}{11}=\frac{42}{11}.$

So, option b is correct.