Multiplying Radicals

Algebra 1 · Axiom Academy

The product rule lets you merge two square roots into one — then a quick factor pulls the perfect square back out. Two separate square roots, multiplied together, become one square root. The numbers under the radicals — the radicands — multiply, and they move under a single radical sign. The index has to match: this works because both are square roots. Multiply what is inside; keep one radical sign The two radicands 6 and 10 slide together into . (We will simplify in Step 3.) Often a number sits in front of the radical, like the 2 in . That number is a coefficient . When you multiply, the coefficients combine with each other out front, and the radicands combine with each other under the sign — two independent lanes. The whole numbers in front join together: . The numbers under the signs join under one radical: . The 2 and 4 stay out front and multiply to 8 ; the 3 and 5 go under one sign and multiply to 15 . Merging often leaves a big radicand that is not yet simplified — like . A square root is fully simplified only when no perfect-square factor is left underneath. So split the radicand into a perfect square times the rest, and pull that square out front as its own root. Find the largest perfect square that divides the radicand. For 60 , that is 4 (since ). Now run both moves in one problem: multiply with the product rule, then simplify by extracting the perfect square. This is the whole skill in a single flow.

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