Like Radicals

Algebra 1 · Axiom Academy

Just as we combine like terms, we can combine like radicals — radical expressions with the same index and the same radicand. 1. A Radical Is a Unit You Can Count Think of as a single object — a unit . Then is just "three of that unit," and is "five of it." Counting them together gives eight of the unit: . It is the identical bookkeeping as combining like terms, where 3x + 5x = 8x . The coefficients are counts; the radical is the shared unit To be alike, a pair must match on both counts — same index and same radicand. Coefficients out front are irrelevant to the test; they are just the counts. If either the index or the radicand differs, the radicals are unlike and cannot be combined as they stand. and — both are square roots of 5 . Like. Combine to . and — both square roots, but of different numbers. Unlike. They stay as . and — same radicand, but a square root versus a cube root. Unlike. and are still alike — only the index and radicand decide. 3. Combining: Add the Coefficients To combine like radicals, add or subtract the coefficients and keep the radical part exactly as it is. This is the distributive property run in reverse: the shared factors out of both terms. 4. Simplify First, Then Combine Some radicals are unlike only on the surface. Before deciding a pair cannot combine, simplify each radical — a hidden perfect-square factor can reveal a common radicand. Take . At a glance the radicands differ, so they look unlike. But , and 4 is a perfect square:

This is the written version of the interactive lesson above. See the full Algebra 1 course.