Simplifying radicals
A square root is exact but hides its size. How do you rewrite expressions with roots so they stay exact and become as simple as they can be?
Where you are 3 inside
What this area is about
An exact answer such as \displaystyle \sqrt{245} can be written more usefully as \displaystyle 7\sqrt{5}. This area is about rewriting square roots and expressions built from them in simplest form, never as rounded decimals. Everything rests on the product rule \displaystyle \sqrt{ab} = \sqrt{a}\,\sqrt{b}, which holds for a \ge 0 and b \ge 0.
How these fit together
All three children are procedures, each using the one before.
Simplifying a square root proves the product rule and pulls perfect square factors out from under the root.
Operations with radicals multiplies and divides roots as one root, and adds or subtracts like radicals such as \displaystyle 3\sqrt{2} + 5\sqrt{2}.
Rationalizing a denominator multiplies top and bottom by the same nonzero number, often a root or a conjugate, so the denominator becomes rational: \displaystyle \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}.
A path through it
Simplifying a square root, then operations with radicals, then rationalizing a denominator.
Next: radical equations, the quadratic formula, whose answers often carry a root, and special right triangles.