To solve the expression [tex]\(\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}}\)[/tex], we can break it down step by step and evaluate the inner square roots iteratively.
Given expression:
[tex]\[
\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}}
\][/tex]
1. Evaluate the innermost square root:
[tex]\[
\sqrt{225} = 15
\][/tex]
2. Substitute and evaluate the next square root:
[tex]\[
\sqrt{154 + 15} = \sqrt{169} = 13
\][/tex]
3. Substitute and evaluate the next square root:
[tex]\[
\sqrt{108 + 13} = \sqrt{121} = 11
\][/tex]
4. Substitute and evaluate the next square root:
[tex]\[
\sqrt{25 + 11} = \sqrt{36} = 6
\][/tex]
5. Substitute and evaluate the final square root:
[tex]\[
\sqrt{10 + 6} = \sqrt{16} = 4
\][/tex]
Therefore, the value of the expression [tex]\(\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}}\)[/tex] is:
[tex]\[
4.0
\][/tex]
