\(\sqrt{3+{\sqrt5}}\) = .........A) √2 + 12.B) \(\sqrt{\cfrac{8}{2

1.	\(\sqrt{3+{\sqrt5}}\) = .........A) √2 + 12.B) \(\sqrt{\cfrac{8}{2}}\) + \(\sqrt{\cfrac{1}{2}}\)C) \(\sqrt{\cfrac{7}{2}}\) - \(\sqrt{\cfrac{1}{2}}\)D) \(\sqrt{\cfrac{9}{2}}\) - \(\sqrt{\cfrac{3}{2}}\)
Answer» Correct option is (C) \(\sqrt\frac{5}{2}+\sqrt\frac{1}{2}\) Let \(\sqrt{3+\sqrt5} \) \(=\sqrt a+\sqrt b\) \(\Rightarrow\) \(3+\sqrt5\) \(=(\sqrt a+\sqrt b)^2\) (By squaring both sides) \(\Rightarrow\) \(a+b+2\sqrt{ab}\) \(=3+\sqrt5\) \(\Rightarrow\) \(a+b+\sqrt{4ab}\) \(=3+\sqrt5\) \(\Rightarrow\) a+b = 3 and 4ab = 5 _______(1) (By comparing rational and irrational parts of both rational numbers) Now, \((a-b)^2=(a+b)^2-4ab\) \(=3^2-5\) = 9 - 5 = 4 \(\therefore\) a - b = 2 (Let) _______(2) From (1) and (2), we obtain (a+b) + (a-b) = 3+2 \(\Rightarrow\) 2a = 5 \(\Rightarrow\) \(a=\frac52\) Then from (1), b = 3 - a \(=3-\frac52=\frac12\) \(\therefore\) \(\sqrt{3+\sqrt5} \) \(=\sqrt a+\sqrt b\) \(=\sqrt{\frac52}+\sqrt{\frac12}\) Correct option is C) \(\sqrt{\cfrac{7}{2}}\) - \(\sqrt{\cfrac{1}{2}}\)

\(\sqrt{3+{\sqrt5}}\) = .........A) √2 + 12.B) \(\sqrt{\cfrac{8}{2}}\) + \(\sqrt{\cfrac{1}{2}}\)C) \(\sqrt{\cfrac{7}{2}}\) - \(\sqrt{\cfrac{1}{2}}\)D) \(\sqrt{\cfrac{9}{2}}\) - \(\sqrt{\cfrac{3}{2}}\)

Answer»

Correct option is (C) \(\sqrt\frac{5}{2}+\sqrt\frac{1}{2}\)

Let \(\sqrt{3+\sqrt5} \) \(=\sqrt a+\sqrt b\)

\(\Rightarrow\) \(3+\sqrt5\) \(=(\sqrt a+\sqrt b)^2\) (By squaring both sides)

\(\Rightarrow\) \(a+b+2\sqrt{ab}\) \(=3+\sqrt5\)

\(\Rightarrow\) \(a+b+\sqrt{4ab}\) \(=3+\sqrt5\)

\(\Rightarrow\) a+b = 3 and 4ab = 5 _______(1) (By comparing rational and irrational parts of both rational numbers)

Now, \((a-b)^2=(a+b)^2-4ab\)

\(=3^2-5\) = 9 - 5 = 4

\(\therefore\) a - b = 2 (Let) _______(2)

From (1) and (2), we obtain

(a+b) + (a-b) = 3+2

\(\Rightarrow\) 2a = 5

\(\Rightarrow\) \(a=\frac52\)

Then from (1), b = 3 - a

\(=3-\frac52=\frac12\)

\(\therefore\) \(\sqrt{3+\sqrt5} \) \(=\sqrt a+\sqrt b\)

\(=\sqrt{\frac52}+\sqrt{\frac12}\)

Correct option is C) \(\sqrt{\cfrac{7}{2}}\) - \(\sqrt{\cfrac{1}{2}}\)