"Here's an expression: \(\sqrt{3+2\sqrt2}\). Can your group find its exact value — no calculator, no decimals? You already have every skill you need to do this. You just don't know it yet."
Expand \((1+\sqrt2)^2\), \((2+\sqrt3)^2\), \((\sqrt2+\sqrt3)^2\).
Compare each result to its starting pair \((m,n)\). What connects them?
Conjecture: \((\sqrt m+\sqrt n)^2 = \) ___ \(+2\sqrt{\text{___}}\)
Find \(m,n\) so that \((\sqrt m+\sqrt n)^2 = 5+2\sqrt6\).
Write two equations linking \(m,n\) to 5 and 6.
Solve for \(m,n\), then write \(\sqrt{5+2\sqrt6}\) as \(\sqrt m+\sqrt n\).
Try \(\sqrt{7+4\sqrt3}\). Rewrite \(4\sqrt3\) as \(2\sqrt{\text{something}}\) first.
Apply the method, then simplify — check if a surd reduces further.
Write the general method for denesting \(\sqrt{a+c\sqrt b}\).
For \(x=\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}\), explain why \(x=\sqrt{1+x}\), then form a quadratic.
Solve with the quadratic formula. Name the number you get.
Replace every 1 with a 2 — what does the nest converge to now?