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