Find $ \sqrt{10+ \sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}} $

Given:

\( \sqrt{10+ \sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}} \)

To do:

We have to find \( \sqrt{10+ \sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}} \)

Solution:

$\sqrt{225}=15$

This implies,

$\sqrt{10+ \sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}}=\sqrt{10+ \sqrt{25+\sqrt{108+\sqrt{154+15}}}}$

$=\sqrt{10+ \sqrt{25+\sqrt{108+\sqrt{169}}}}$

$=\sqrt{10+ \sqrt{25+\sqrt{108+13}}}$            [$\sqrt{169}=13$]
$=\sqrt{10+ \sqrt{25+\sqrt{121}}}$

$=\sqrt{10+ \sqrt{25+11}}$            [$\sqrt{121}=11$]

$=\sqrt{10+ \sqrt{36}}$

$=\sqrt{10+ 6}$            [$\sqrt{36}=6$]

$=\sqrt{16}$

$=4$            [$\sqrt{16}=4$]

Hence, $\sqrt{10+ \sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}}=4$.

Tutorialspoint

Updated on: 10-Oct-2022

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