Question

If $y=\sqrt{\sqrt{7}+7+\sqrt{8+2\sqrt{7}}}-\sqrt{7}$, then find the value of $y$. [3 marks]

Answer 1

We see that,
$y=\sqrt{\sqrt{7}+7+\sqrt{8+2\sqrt{7}}}-\sqrt{7}$
Here, $8+2\sqrt{7}=7+1+2\sqrt{7}=(\sqrt{7}+1{)}^2$
On solving for $y$, we get
$\left[1mark\right]$
$y=\sqrt{\sqrt{7}+7+\sqrt{(\sqrt{7}+1{)}^2}}-\sqrt{7}$
$y=\sqrt{\sqrt{7}+7+(\sqrt{7}+1)}-\sqrt{7}$
$y=\sqrt{2\sqrt{7}+7+1}-\sqrt{7}$
Using identity, we get
$\left[1mark\right]$
$y=\sqrt{(\sqrt{7}+1{)}^2}-\sqrt{7}$
$y=(\sqrt{7}+1)-\sqrt{7}$
$\therefore y=1$
$\left[1mark\right]$