A Radical Departure...
(Inspired by Ramanujan and an excellent Wikipedia article on Nested Radicals)
Suggestion: Assign as a 2-day team or individual project after demonstrating a similar but simpler example such as the square root of 3+2√2 = 1+√2.
NOTE: The method below DOES NOT show a detailed algebraic solution, using substitutions and solution of resulting quadratic equations. Rather, I suggested some reasonable educated guessing, aka number sense. I would recommend both approaches.
There is considerable more theory than is suggested by this example, e.g., justification of uniqueness of roots, conditions for roots to be of the form suggested in the solution, etc. Encourage students to investigate further!
PROBLEM: Demonstrate the following identity by simplification of the left-hand side only. No calculators permitted for derivation although numerical (decimal) verification that the left side equals the right is recommended prior to starting the 'proof'.
(SOLUTION GIVEN BELOW STATEMENT OF IDENTITY)
NOTE: Illegibility of next to last line of 3rd image! Should be (Square root of 3 + Square of 2) not 'Square root of 4'.
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