Root Calculator

How the root calculator works

The nth root of a number x is the number r with rⁿ = x. The square root (n = 2) of 81 is 9 because 9² = 81; the cube root (n = 3) of 125 is 5 because 5³ = 125. Choose square, cube or nth, type the number under the root sign (the radicand) and you get:

• the root as a decimal, or exactly when the number is a perfect power (√144 = 12, ∛0.001 = 0.1);
• for whole numbers, the simplified radical form, such as √72 = 6√2, and the prime factorization it comes from;
• the nearest perfect squares, cubes or nth powers below and above your number;
• for negative numbers, the real root when n is odd, or the principal complex root when n is even.

For roots inside longer expressions, use the scientific calculator; to find which power of a base gives a number (the opposite question), use the log calculator.

Roots, powers and simplifying radicals

x

the radicand, the number under the root sign

n

the index: 2 for a square root, 3 for a cube root

Simplifying. Split the radicand into primes and pull every complete group of n equal factors out of the root. Each group of n copies of a prime p leaves the root as a single p:

The result is in simplest form when no prime appears n or more times in m. If m is 1, the number is a perfect nth power and the root is the whole number c.

Negative radicands. An odd power keeps the sign, so an odd root of a negative number is negative: ∛(−27) = −3 because (−3)³ = −27. An even power is never negative, so an even root of a negative number has no real value. It does have complex values; the principal one (the standard choice) for a square root is √(−a) = i√a, where i² = −1. For a higher even index n the principal root has the size ⁿ√|x| and points at the angle 180°/n: ⁿ√|x| × (cos(180°/n) + i sin(180°/n)).

Worked example: √72

1. Prime factorization: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3².
2. Pull out pairs (n = 2): one pair of 2s and one pair of 3s come out, one 2 stays inside. So 72 = (2 × 3)² × 2 = 6² × 2.
3. √72 = √(6² × 2) = 6√2.
4. As a decimal: 6 × 1.4142135624 ≈ 8.4852813742.
5. Check against the nearest perfect squares: 64 = 8² and 81 = 9², so √72 must lie between 8 and 9.

A cube root. 54 = 2 × 3³, so ∛54 = 3∛2 ≈ 3.7797631497. The nearest perfect cubes are 27 = 3³ and 64 = 4³. For −54 the same working gives ∛(−54) = −3∛2.

A negative square root. √(−16) = √16 × √(−1) = 4i. Squaring checks it: (4i)² = 16 × i² = −16.

Tips

• Estimate first. The nearest perfect powers bracket the answer. √50 lies just above 7 because 49 = 7², which is a quick check on any result.
• Roots of fractions and decimals. Take the root of the top and the bottom: √0.25 = √(1/4) = 1/2 = 0.5. A decimal has an exact root only when both parts of its fraction are perfect powers.
• Roots in geometry. The length of a right triangle's long side is a square root, c = √(a² + b²); the Pythagorean theorem calculator works it through, and the quadratic equation solver gives roots of equations in the same simplified radical form.

Limits

The radicand can be any number from −10¹⁵ to 10¹⁵ and the index a whole number from 2 to 100. The simplified radical form and the prime factorization are given for whole numbers; for other decimals you get the decimal root, or the exact value when the decimal is a perfect power. Decimals are shown to about 10 digits. For negative numbers with an even index the calculator shows only the principal complex root, not all n of them.