Square Root Calculator

The square root of a non-negative number x is a value r such that r multiplied by itself equals x, represented mathematically as r² = x. When computing square roots manually or programmatically, the tool determines both the positive principal root and evaluates whether the input is a perfect square. If the number is not a perfect square, the calculator breaks the radicand down into its simplest radical form by extracting the largest perfect square factor, ensuring your mathematical expressions remain exact and fully reduced.

Let us find the square root and simplified radical form of the number 75. First, we identify the largest perfect square that divides evenly into 75, which is 25. Next, we rewrite the expression as the product of two square roots: √(75) = √(25 × 3). Using the multiplication property of radicals, this separates into √25 × √3. Because the square root of 25 is exactly 5, our final simplified exact answer is 5√3, which evaluates numerically to approximately 8.660.

When working with square roots in algebra, always try to simplify radicals rather than converting them to long decimals immediately to preserve precision. Remember that every positive real number has both a positive and a negative square root, though the radical symbol specifically denotes the principal non-negative root. If you are estimating square roots by hand, bound your target number between the two closest adjacent perfect squares to narrow down your decimal approximation quickly.