Root Calculator

About the Root Calculator

A root calculator computes square roots, cube roots, nth roots, and fractional roots of any positive or negative number — including both decimal and exact simplified radical form. Roots are the inverse operation of powers: the square root of 9 is 3 because 3^2 = 9; the cube root of -27 is -3 because (-3)^3 = -27. Roots appear throughout mathematics, physics, engineering, and statistics: the Pythagorean theorem uses square roots; the quadratic formula has a square root; standard deviation is the square root of variance; RMS voltage is a root-mean-square calculation; and the nth root appears in compound interest formulas when solving for rate. Our calculator handles all real roots including negative radicands for odd roots, displays simplified radical form (√72 = 6√2), and shows step-by-step simplification using prime factorisation to help students learn the process.

How It Works

Square root: sqrt(x) = x^(1/2). Cube root: cbrt(x) = x^(1/3). Nth root: n-th-root(x) = x^(1/n). Negative base rules: even roots of negatives produce imaginary numbers (sqrt(-4) = 2i — not real); odd roots of negatives are real (cbrt(-27) = -3). Simplifying radicals: factor out perfect square factors. sqrt(72) = sqrt(36 x 2) = sqrt(36) x sqrt(2) = 6sqrt(2). sqrt(180) = sqrt(4 x 9 x 5) = 2 x 3 x sqrt(5) = 6sqrt(5). Rationalising denominators: 1/sqrt(2) = sqrt(2)/2 (multiply numerator and denominator by sqrt(2)). Fractional exponents: x^(m/n) = n-th-root(x^m) = (n-th-root(x))^m. 8^(2/3) = (cbrt(8))^2 = 2^2 = 4.

Tips & Best Practices

• ✓Perfect squares to memorise: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400.
• ✓sqrt(2) ≈ 1.41421 and sqrt(3) ≈ 1.73205 appear constantly in geometry — the diagonal of a unit square and the height of an equilateral triangle respectively.
• ✓RMS (Root Mean Square): the RMS of AC voltage = sqrt(average of squared instantaneous voltages). For a sinusoidal AC wave: RMS = peak voltage / sqrt(2). A 120V RMS outlet has a peak voltage of 120 x sqrt(2) ≈ 170V.
• ✓The irrational nature of sqrt(2): proven irrational by the ancient Greeks — it cannot be expressed as any fraction p/q where both are integers. Its decimal expansion is infinite and non-repeating.
• ✓Compound interest rate solving: to find the annual rate r that doubles money in n years: r = 2^(1/n) - 1. To double in 10 years: r = 2^(1/10) - 1 = 1.0718 - 1 = 7.18% per year.
• ✓Simplification strategy: find the largest perfect square factor, then use the product rule. For sqrt(1,008): 1,008 = 16 x 63 = 16 x 9 x 7. sqrt(1,008) = 4 x 3 x sqrt(7) = 12sqrt(7).
• ✓Complex roots: when a quadratic has a negative discriminant, its roots are complex conjugates involving sqrt(-1) = i. These are valid mathematical objects used extensively in electrical engineering and signal processing.
• ✓Newton's method for computing roots: iterative approximation formula x_{n+1} = (1/n) x ((n-1)x_n + a/x_n^(n-1)) converges rapidly to the nth root of a. This is how calculators actually compute roots internally.

Who Uses This Calculator

Algebra and geometry students simplifying radical expressions and solving equations involving roots. Physics students computing RMS values and solving kinematic equations. Statistics students computing standard deviations from variance values. Engineers using nth root formulas for compound interest rate calculations. Chemistry students solving equilibrium expressions. Teachers creating worked examples of radical simplification for coursework.