Radical Expressions & Rationalization

Calculus Readiness · Axiom Academy

Simplify square roots, combine like radicals, rationalize denominators — and see why a radical is really just a fractional exponent. 1. Pull the Perfect Square Out To simplify a square root, split the radicand into a perfect square times whatever is left. The product rule then lets the square walk out of the radical as a whole number, while the leftover stays inside. the perfect square 36 becomes 6 and escapes the leftover 2 has no square factor — it stays in Same move with letters: , where 25x^2 is the perfect square. You can add or subtract radicals only when they match — same index and same radicand. Treat like a variable: behaves exactly like 3y + y . The trick is to simplify first , because two roots that look different often hide the same radical. — just add the coefficients; the rides along. does not combine. Different radicands — leave it as is. — radicands multiply, then simplify. Multiplying binomials — use FOIL Radical binomials multiply just like algebra. For , expand and use : 3. Rationalize the Denominator By convention we don't leave a radical in the denominator. To clear it, multiply by a clever form of 1 : the radical over itself. Since , the radical below squares into a whole number and effectively lifts up to the numerator. Monomial denominator: multiply top and bottom by the radical itself, as in the animation: . Binomial denominator: multiply by the conjugate — flip only the middle sign. The difference of squares (a+b)(a-b)=a^2-b^2 kills the radical.

This is the written version of the interactive lesson above. See the full Calculus Readiness course.