Simplifying Radicals Calculator

A radical expression is in its simplest form when the radicand (the number under the radical sign) contains no factor that is a perfect square (for square roots). The mathematical foundation relies on the product property of radicals: √(a × b) = √a × √b, provided that a and b are non-negative real numbers. To simplify a radical like √N, the calculator finds the largest perfect square factor A^2 that divides evenly into N, allowing it to rewrite the expression as A√B.

Let us walk through simplifying the radical expression √72 step by step. First, identify the factors of 72 to find the largest perfect square that divides evenly into it. The factors of 72 include 1, 4, 8, 9, 12, 18, 24, 36, and 72. Among these, 36 is the largest perfect square. We can rewrite the expression using the product property: √72 = √(36 × 2). Next, split the radical into two separate roots: √36 × √2. Since the square root of 36 is exactly 6, the fully simplified expression becomes 6√2.

When simplifying radicals by hand, always begin by finding prime factorizations if you cannot immediately spot large perfect squares. Group identical prime factors into pairs for square roots, moving one factor from each pair outside the radical sign. Be cautious when adding or subtracting radical expressions; you can only combine terms that share both the exact same index and the exact same radicand, much like combining like terms in basic algebra.