Question

Choose the correct answer.

• A. $\sqrt{2-\sqrt{3}}>\sqrt{\sqrt{7}-\sqrt{5}}$
• B. $\sqrt{2-\sqrt{3}}<\sqrt{\sqrt{7}-\sqrt{5}}$
• C. $\sqrt{\sqrt{7}-\sqrt{5}}>\sqrt{\sqrt{2}-1}$
• D. $\sqrt{\sqrt{2}-1}<\sqrt{2-\sqrt{3}}$

Answer 1

The correct option is B $\sqrt{2-\sqrt{3}}<\sqrt{\sqrt{7}-\sqrt{5}}$

Let us compare $\sqrt{\sqrt{2}-1},\sqrt{2-\sqrt{3}}$ and $\sqrt{\sqrt{7}-\sqrt{5}}.$

Approximate values:
$\sqrt{2}=1.414,\sqrt{3}=1.732,$
$\sqrt{5}=2.236,\sqrt{7}=2.645$

$\therefore \sqrt{\sqrt{2}-1}=\sqrt{1.414-1}=\sqrt{0.414}$

$\sqrt{2-\sqrt{3}}=\sqrt{2-1.732}=\sqrt{0.268}$

$\sqrt{\sqrt{7}-\sqrt{5}}=\sqrt{2.645-2.236}=\sqrt{0.409}$

$\because 0.268<0.409<0.414,$

$\therefore \sqrt{0.268}<\sqrt{0.409}<\sqrt{0.414}.$

$⟹\sqrt{2-\sqrt{3}}<\sqrt{\sqrt{7}-\sqrt{5}}<\sqrt{\sqrt{2}-1}$

Thus, the correct answer is $\sqrt{2-\sqrt{3}}<\sqrt{\sqrt{7}-\sqrt{5}}.$