The expression sqrt(20 - sqrt(20 - sqrt(20 - sqrt(20 - \dots)))) is equal to ________.

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Accepted Answer

{"content":[{"content":[{"marks":[{"type":"bold"}],"text":"Step 1 (Formulate infinite nested radical equation):","type":"text"},{"text":" Let the infinite nested radical expression be represented by ","type":"text"},{"attrs":{"latex":"y"},"type":"math_inline"},{"text":":","type":"text"}],"type":"paragraph"},{"type":"math_block","attrs":{"latex":"y = \sqrt{20 - \sqrt{20 - \sqrt{20 - \dots}}}"}},{"content":[{"text":"Step 2 (Identify self-similar term):","type":"text","marks":[{"type":"bold"}]},{"text":" Since the nested radical is infinite, the expression under the outermost square root is itself equal to ","type":"text"},{"attrs":{"latex":"y"},"type":"math_inline"},{"text":":","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"y = \sqrt{20 - y}"},"type":"math_block"},{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 3 (Solve the quadratic equation):","type":"text"},{"text":" Square both sides:","type":"text"}]},{"attrs":{"latex":"y^2 = 20 - y"},"type":"math_block"},{"attrs":{"latex":"y^2 + y - 20 = 0"},"type":"math_block"},{"type":"math_block","attrs":{"latex":"(y + 5)(y - 4) = 0"}},{"content":[{"text":"This gives two solutions ","type":"text"},{"attrs":{"latex":"y = -5"},"type":"math_inline"},{"text":" and ","type":"text"},{"attrs":{"latex":"y = 4"},"type":"math_inline"},{"text":". By definition, the principal square root is non-negative (","type":"text"},{"attrs":{"latex":"y \ge 0"},"type":"math_inline"},{"text":"). Thus, we discard ","type":"text"},{"attrs":{"latex":"y = -5"},"type":"math_inline"},{"text":".","type":"text"}],"type":"paragraph"},{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 4 (Conclusion):","type":"text"},{"text":" The expression is exactly equal to ","type":"text"},{"marks":[{"type":"code"}],"text":"4","type":"text"},{"text":".","type":"text"}]}],"type":"doc"}

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