Racine Carrée Calculator

The radical sign √ (from the Latin radix, “root”) groups the number or expression underneath—the radicand. Writing √25 asks for the non-negative number whose square is 25.

In typesetting and on calculators you may also see sqrt(x) or x^(1/2). All mean the same principal square root when x ≥ 0 in real arithmetic.

Squaring a number means multiplying it by itself (3² = 3 × 3 = 9). The square root reverses that process: it finds the side length of a square with a given area.

In French the same idea is racine carrée—“square root”—because geometry and algebra both tie √x to squares. A square of area x has side length √x.

For a non-negative real number x, the principal square root √x is the unique real number y ≥ 0 such that y² = x.

Examples: √0 = 0, √1 = 1, √9 = 3, and √2 is irrational (about 1.414…). Every positive real has exactly one principal square root.

The notation √(a²) = |a| for all real a reminds us that square roots undo squaring but always return a non-negative value in the principal branch.

Methods depend on whether x is a perfect square or not:

• Perfect squares (1, 4, 9, 16, 25, …): recall the integer whose square matches—√144 = 12.
• Factorization: split x into a perfect square times another factor—√72 = √(36×2) = 6√2.
• Estimation: bracket between nearby squares; √50 is between √49 = 7 and √64 = 8.
• Calculator or computer: use a square-root key, this page’s tool, or a spreadsheet function such as SQRT(x).

Enter any non-negative x in the workspace above. The tool returns a decimal approximation and, for whole-number inputs, a simplified radical when possible (e.g. √50 → 5√2).

Leave negative inputs out—real square roots are not defined for x < 0. Use complex numbers separately if your course requires imaginary results.

Jump to the calculator ↑

To simplify √n, factor n and pull out the largest perfect-square divisor.

Example: √200 = √(100×2) = 10√2. The radicand under the sign (2) should have no perfect-square factors other than 1—that is called simplest radical form.

• List prime factors or factor pairs of n.
• Pair equal factors; each pair becomes one factor outside the radical.
• Multiply outside factors and leave the rest inside—√(18) = √(9×2) = 3√2.

Radicals combine only when the radicand matches (like terms):

• Addition / subtraction: 3√5 + 2√5 = 5√5, but 3√5 + 2√3 cannot merge further.
• Multiplication: √a · √b = √(ab)—e.g. √2 · √8 = √16 = 4.
• Division: √a / √b = √(a/b) when b > 0; rationalize denominators when teaching requires it.
• Distribute carefully: 2(√3 + √2) = 2√3 + 2√2.

Always simplify each radical first so like terms are easy to spot.

Exponent rules link roots to powers: √x = x^(1/2) for x ≥ 0. Therefore √(x²) = |x|, not always x.

For fractions: √(a/b) = √a / √b when a ≥ 0 and b > 0. Example: √(9/16) = 3/4.

Nested roots follow the same idea: √(√16) = √4 = 2. Powers inside combine: √(a²b) = |a|√b for real a, b ≥ 0.

The function f(x) = √x is defined for x ≥ 0. Its graph starts at the origin, rises slowly, and is always increasing—each step in x adds less vertical change as x grows.

The curve is half of a sideways parabola. It is concave down: doubling x does not double √x (√4 = 2 but √8 ≈ 2.83, not 4).

For f(x) = √x = x^(1/2) with x > 0, calculus gives f′(x) = 1 / (2√x).

The slope is steepest near zero and flattens as x increases—matching the graph’s shape. At x = 4, f′(4) = 1/4.

Real numbers cannot square to a negative value, so √−4 has no real answer. This calculator stops with an error for negative input.

In complex analysis, √−1 is defined as i (the imaginary unit), and √−9 = 3i in the principal branch—but that is a separate number system from real square roots on homework checks.