Rationalize denominators, solve radical equations, and apply the distance formula.
By convention, a "finished" answer never leaves a radical sitting in the denominator. To rationalize it, multiply the fraction by a clever form of 1 - the radical over itself - so 1/√3 becomes (1/√3)·(√3/√3) = √3/3. The value never changes (you only multiplied by 1), but the radical has moved to the top, where convention says it belongs.
A radical equation is solved by isolating the radical, then squaring both sides to undo it - but squaring is a one-way door: it can turn a false equation into one that LOOKS true, creating an extraneous solution that must be caught by checking in the ORIGINAL equation, not the squared version. This same square-root machinery gives you the distance formula, d = √((x₂-x₁)² + (y₂-y₁)²) - nothing more than the Pythagorean theorem wearing coordinates, with the two legs built from the x-difference and y-difference between two points.
Mastery looks like: They rationalize denominators, solve radical equations while correctly checking for extraneous solutions, and apply the distance formula.
Common stumbles: Forgetting to check for extraneous solutions after squaring; rationalizing by multiplying only the denominator instead of the whole fraction by √a/√a.