Let's work with the expression "The fourth power of $\sqrt{1+\sqrt{1+\sqrt{1}}}$" (from Salkind, 1970). If you were asked to simplify this algebraically, what would you get?

Now, program this formula into the computer, so it'll output the numerical value of this expression (to the fourth power). Next, program in your simplified algebraic answer. Are the two outputs the same (they should be!).

This problem should show you the theme or power of using a computer to do math, in particular, that in some cases, you don't need the algebra! Just let the computer crunch the numbers!

When you're done with the above expression, do the same for $\sqrt{3+2\sqrt{2}}-\sqrt{3-2\sqrt{2}}$.