Question

Please solve:- $\sqrt{ \frac{81}{64} \sqrt{ \frac{81}{64} \sqrt{ \frac{81}{64} \sqrt{ \frac{81}{64} } }}}.................. \infty}$

AND
$\sqrt{10+ \sqrt{24} + \sqrt{60} + \sqrt{40} }$

Answer 1

$x = \frac{81}{64} \\ \\ Given\ expr\ E = \sqrt{x \sqrt {x\sqrt{x\ ...}}} \\ \\ E = \sqrt{x E} \ \ \ \ \ So,\ \ \ E^2 = x E \ \ \ \ E = x = \frac{81}{64}\ OR,\ another\ way, \\ \\ E = x^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}...} = x^1 = x = \frac{81}{64} \\ \\ \\ Y = \sqrt{10+\sqrt{24}+\sqrt{60}+\sqrt{40}} \\ \\ Y^2 = 10 + \sqrt{24}+\sqrt{60}+\sqrt{40}$