Square and cube roots
You know the area of a square or the volume of a cube. How do you get back to the length of its side?
Where you are 3 inside
What this area is about
Squaring and cubing take a length to an area or a volume. A root runs that backwards: b is a square root of a when b^2 = a, and a cube root of a when b^3 = a. Much of the area is about how many answers there are, and when there are none.
How these fit together
Square roots. Square roots reads roots straight off the perfect squares, and fractions and decimals built from them. A positive number has two square roots, one positive and one negative; \displaystyle \sqrt{a} names the one that is not negative, and a negative number has no real square root.
Cube roots. Cube roots and higher roots undoes cubing. Every real number has exactly one real cube root, so a negative radicand is allowed: \displaystyle \sqrt[3]{-64} = -4.
Any index. Higher roots treats \displaystyle \sqrt[n]{a} for whole numbers n \ge 2. Whether n is odd or even decides almost everything: an odd index gives exactly one real root for every real a, while an even index gives a real root only when a \ge 0.
A path through it
Square roots first, then cube roots, then higher roots: each builds on the one before.
Finish here and you can simplify a square root, see why \displaystyle \sqrt{2} is irrational, use the Pythagorean theorem, solve quadratics with square roots and read rational exponents such as a^{1/2}.