Simplify, combine, and rationalize expressions involving radicals.
A radical is simplified when no perfect-square factor (or perfect-cube, for cube roots) is left hiding inside it. Pull out any perfect square factor you can find, the same way you would simplify a fraction by pulling out a common factor. Radicals can only be combined by addition or subtraction when they are "like radicals" -- the exact same number under the exact same root -- just like combining like terms.
A denominator is not considered "simplified" while it still contains a radical, so we rationalize it: multiply top and bottom by whatever clears the root from the bottom. For a single-term radical denominator, multiply by that same radical over itself. For a two-term denominator like (2+sqrt3), multiply by its conjugate (2-sqrt3) instead, since (a+b)(a-b)=a^2-b^2 eliminates the root entirely.
Mastery looks like: They simplify radicals by extracting perfect-square factors, combine like radicals, and rationalize both single-term and two-term radical denominators.
Common stumbles: Combining unlike radicals as if they were like terms; rationalizing a two-term denominator by multiplying by the same expression instead of its conjugate.
See it visually
Interactive, spoken-aloud tools that show this idea in motion.