Square Root Calculator

A square root of a number x is a value y such that y × y = x. For example, the square root of 25 is 5 because 5 × 5 = 25. Every positive number has two square roots: a positive and a negative one (both 5 and −5 squared give 25). By convention, the "square root" refers to the principal (positive) root, written as √25 = 5.

A perfect square is a number whose square root is a whole integer. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Non-perfect squares like 2, 3, 5, 7 have irrational square roots — they cannot be expressed as a fraction and their decimal form never ends or repeats. √2 = 1.41421356... continues infinitely.

To simplify √72: find the largest perfect square that divides 72. That is 36 (since 36 × 2 = 72). Then: √72 = √(36 × 2) = √36 × √2 = 6√2. The simplified form keeps the perfect square factor outside the radical sign and the remaining factor inside. This calculator shows the simplified form automatically for integer inputs.

√2 = 1.41421356237... It is irrational, meaning its decimal never ends or repeats. This was one of the first numbers proved irrational, by the ancient Greeks. It appears everywhere in geometry: the diagonal of a 1×1 square has length √2. It also appears in the ISO 216 paper size standard: A4 paper has a ratio of 1:√2 so that folding it in half gives the same proportions.

Not in real numbers. The square root of a negative number is imaginary. In complex number math, √(−1) = i (the imaginary unit). So √(−9) = 3i. Imaginary numbers are used in electrical engineering, quantum mechanics, control theory, and signal processing. This calculator works with non-negative real numbers only.

The cube root of x is the number y such that y × y × y = x. Written as ∛x or x^(1/3). Examples: ∛8 = 2 (2×2×2=8). ∛27 = 3 (3×3×3=27). ∛125 = 5. Unlike square roots, cube roots exist for negative numbers: ∛(−8) = −2 ((−2)×(−2)×(−2) = −8). Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

The nth root of x (written ⁿ√x or x^(1/n)) is the number y such that y^n = x. Square root = 2nd root. Cube root = 3rd root. 4th root of 81 = 3 (3^4 = 81). 5th root of 32 = 2 (2^5 = 32). Fractional exponents express roots: x^(1/n) = ⁿ√x. More generally, x^(m/n) = (ⁿ√x)^m. This is how calculators compute any root using the power function.

√0 = 0 (because 0 × 0 = 0). √1 = 1 (because 1 × 1 = 1). Both are perfect squares. √1 is also notable because 1 is the only positive number whose square root equals itself (other than conceptually treating 0 as its own square root). For any n, the nth root of 0 = 0 and the nth root of 1 = 1.

Find the two perfect squares it falls between. Example: √50. Since 49 = 7² and 64 = 8², √50 is between 7 and 8, closer to 7. Better estimate: 50 is 1/15 of the way from 49 to 64, so √50 ≈ 7 + 1/15 ≈ 7.07. Actual: 7.0711. Another method: the Babylonian method (Newton’s method) repeatedly applies x_new = (x + n/x) / 2 starting from a guess, converging quickly to the answer.

√x = x^(1/2). This is because x^(1/2) × x^(1/2) = x^(1/2+1/2) = x^1 = x. More generally, ⁿ√x = x^(1/n). Square root laws: √(ab) = √a × √b. √(a/b) = √a / √b. (√a)² = a. These allow simplification of complex radical expressions. Note: √(a+b) ≠ √a + √b — this is a common error.