Question

$\sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + \sqrt{225} } } } }$
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Answer 1

Answer:

4

Step-by-step explanation:

$\sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + \sqrt{225} } } } } \\ \\ =\sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + 15 } } } } \\ \\ = \sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{169} } } } \\ \\ = \sqrt{10 + \sqrt{25 + \sqrt{108 + 13 } } } \\ \\ = \sqrt{10 + \sqrt{25 + \sqrt{121 } } } \\ \\ = \sqrt{10 + \sqrt{25 + 11 } } \\ \\ = \sqrt{10 + \sqrt{36 } } \\ \\ = \sqrt{10 + 6 } \\ \\ = \sqrt{16 } \\ = \sqrt{2 \times 2 \times 2 \times 2 } \\ = 2 \times 2 \\ = 4$

Hence, we can conclude that the final answer or the final result is 4.

Answer 2

Answer:

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

Step-by-step explanation:

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

$\sqrt{225}$

$\mathrm{Factor\:the\:number:\:}\:225=15^2$

$=\sqrt{15^2}$

$\mathrm{Apply\:radical\:rule}:\quad \sqrt[n]{a^n}=a$

$\sqrt{15^2}=15$

$=\sqrt{154+15}$

$\mathrm{Add\:the\:numbers:}\:154+15=169$

$=\sqrt{169}$

$\mathrm{Factor\:the\:number:\:}\:169=13^2$

$=\sqrt{13^2}$

$\mathrm{Apply\:radical\:rule}:\quad \sqrt[n]{a^n}=a$

$\sqrt{13^2}=13$

$=\sqrt{108+13}$

$\mathrm{Add\:the\:numbers:}\:25+11=36$

$=\sqrt{36}$

$\mathrm{Factor\:the\:number:\:}\:36=6^2$

$=\sqrt{6^2}$

$\mathrm{Apply\:radical\:rule}:\quad \sqrt[n]{a^n}=a$

$\sqrt{6^2}=6$

$=\sqrt{10+6}$

$\mathrm{Add\:the\:numbers:}\:10+6=16$

$=\sqrt{16}$

$\mathrm{Factor\:the\:number:\:}\:16=4^2$

$=\sqrt{4^2}$

$\mathrm{Apply\:radical\:rule}:\quad \sqrt[n]{a^n}=a$

$\sqrt{4^2}=4$

$=4$