Radicals All the Way Down

Now try going the other way. You're given the target expression \(5+2\sqrt6\), and you want to find \(m\) and \(n\) such that \((\sqrt{m}+\sqrt{n})^2\) equals it exactly.

Using your Phase 1 rule, write two equations connecting \(m\) and \(n\) to the numbers 5 and 6.

Find whole numbers \(m\) and \(n\) that satisfy both equations at once. (Try a few small pairs.)

Use your values of \(m\) and \(n\) to write \(\sqrt{5+2\sqrt6}\) in the simplified form \(\sqrt{m}+\sqrt{n}\).

Try your method on \(\sqrt{7+4\sqrt3}\). Something's different this time — the coefficient in front of the surd is 4, not 2. Before your Phase 2 method will work, rewrite \(4\sqrt3\) in the form \(2\sqrt{\text{something}}\).

You should now have an expression that looks like \(7+2\sqrt{\text{?}}\). Find \(m\) and \(n\) the same way you did in Phase 2.

Write \(\sqrt{7+4\sqrt3}\) in the form \(\sqrt{m}+\sqrt{n}\) — then check whether one of those surds simplifies further.

In your own words, write down the complete method for denesting any expression of the form \(\sqrt{a+c\sqrt{b}}\) — from spotting a coefficient problem through to the final simplified answer.

Consider this radical, continuing forever:

\(x = \sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}\)

Look carefully at what's inside the very first root. Explain to your group why we're allowed to write:

\(x = \sqrt{1+x}\)

Square both sides, then rearrange everything onto one side to form a quadratic equation equal to zero.

Use the quadratic formula to solve your equation.

You'll get two solutions. Since \(x\) is a length (built from square roots), only one of them makes sense — which one, and why?

Find the decimal value of your answer. Does this number look familiar from anywhere else — art, architecture, nature?