Root Calculator

This root calculator handles any root you can throw at it. Use it as a square root calculator, a cube root calculator, a fourth root calculator, or a general nth root calculator (also called a radical calculator). Enter the number under the radical (the radicand) and the root index n, and the tool returns the principal real root along with a verification check and a perfect-power flag.

1. Enter the radicand. This is the number you want the root of. Type 64, 2, 0.25, or any real value. Negative numbers are accepted when the root index is odd (cube root, 5th root, 7th root).
2. Enter the root index (n). Use 2 for a square root, 3 for a cube root, 4 for a fourth root, or any integer up to whatever you need. The preset buttons load n = 2, 3, or 4 in one click.
3. Read the result. The big number is the principal nth root. The verification row raises that result back to the nth power and should match your original radicand within floating point precision. The perfect-power row tells you whether the radicand is a clean integer power.
4. Scan the comparison card. The bottom panel shows the square root, cube root, and fourth root of the same input side by side, which is handy when you are simplifying radicals or checking which roots come out to integers.

The output is the principal (non-negative) real root. Every positive number also has a negative square root (for example √64 = ±8), and complex nth roots exist in the complex plane, but this calculator returns the one real principal value that answers most practical questions.

nth root of a = a^(1/n)

Special cases:
  Square root (n=2): √a = a^(1/2)
  Cube root (n=3):   ∛a = a^(1/3)
  4th root (n=4):    ⁴√a = a^(1/4)

Examples:
  √64   = 64^(1/2) = 8    (8² = 64, perfect square)
  ∛64   = 64^(1/3) = 4    (4³ = 64, perfect cube)
  ⁴√64  = 64^(1/4) ≈ 2.828 (not a perfect 4th power)
  ⁴√256 = 256^(1/4) = 4   (4⁴ = 256, perfect 4th power)

Definition of the nth root

The nth root of a number a is the number b such that bⁿ = a. It is written as ⁿ√a or as the rational exponent a^(1/n). Both notations mean the same thing. For example, ⁴√81 = 3 because 3⁴ = 81, and 81^(1/4) = 3 says the same thing in exponent form. The number a under the radical sign is called the radicand, and n is called the index or root index.

Square root, cube root, and nth root

When n = 2 we drop the index and just write √a (the square root). When n = 3 we write ∛a (the cube root). For every integer n ≥ 2 the symbol ⁿ√a is defined. There is no reason the index has to stop at 4: the 10th root of 1024 is exactly 2, because 2¹⁰ = 1024, and the 12th root of 4096 is also 2. The calculator above accepts any positive integer n.

Rational exponent form

Rational exponents unify powers and roots into one rule: a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ. Both orderings give the same answer when a is non-negative. For example, 8^(2/3) = ∛(8²) = ∛64 = 4, and also (∛8)² = 2² = 4. This is why scientific calculators treat √ as nothing more than the ^0.5 key, and why a financial formula like CAGR can use either notation interchangeably.

Negative radicands

Odd roots of negative numbers are real and negative. ∛(−8) = −2 because (−2)³ = −8. The 5th root of −32 is −2 because (−2)⁵ = −32. Even roots of negative numbers are not real. √(−4) has no real answer because any real number squared is non-negative. In the complex number system √(−4) = 2i, and in general √(−a) = i√a for a > 0. This calculator stays in the reals and flags negative radicand with even index as not real.

Newton's method for computing √N

Newton's method (also called the Babylonian method for square roots) converges to √N very quickly using the recurrence x_{k+1} = ½(x_k + N ÷ x_k). Start with any positive guess and iterate. For √10 starting at x₀ = 3:

x₀ = 3
x₁ = ½(3 + 10 ÷ 3)       = ½(3 + 3.33333) = 3.16667
x₂ = ½(3.16667 + 10 ÷ 3.16667) = ½(3.16667 + 3.15789) = 3.16228
x₃ = ½(3.16228 + 10 ÷ 3.16228) ≈ 3.16227766

True value: √10 ≈ 3.16227766...

Three iterations from a crude guess produced eight correct digits. The number of correct digits roughly doubles each step, which is called quadratic convergence. The same idea generalizes to nth roots with x_{k+1} = ((n − 1) × x_k + N ÷ x_k^(n−1)) ÷ n.

Simplifying radicals and rationalizing the denominator

A radical is in simplest form when no perfect nth-power factor remains under the radical. Pull those factors out: √72 = √(36 × 2) = 6√2, and √200 = √(100 × 2) = 10√2. For cube roots: ∛54 = ∛(27 × 2) = 3∛2. Rationalizing the denominator means removing radicals from the bottom of a fraction. For 1 ÷ √2 multiply top and bottom by √2 to get √2 ÷ 2. For 1 ÷ (3 − √5) multiply by the conjugate (3 + √5) ÷ (3 + √5) to get (3 + √5) ÷ 4.

Quick reference table

Roots show up everywhere once you start looking. The square root sits inside the Pythagorean theorem, the standard deviation, the root mean square, and the quadratic formula. Cube roots and higher roots appear in volume scaling, compound growth rates, and signal processing. A good nth root calculator or radical calculator shortens any of these jobs from a page of arithmetic to a single lookup.

Where roots actually get used

In geometry the distance between two points (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²), so every map, GPS fix, and collision check in a game engine calls a square root. In physics the RMS (root mean square) voltage of a sine wave is the peak divided by √2, and the standard deviation of a dataset is the square root of the variance. In finance, the compound annual growth rate from a value V₀ to Vₙ over n years is CAGR = (Vₙ ÷ V₀)^(1/n) − 1, which is an nth root in disguise. In cryptography, RSA decryption is essentially taking an eth root mod N, which is easy if you know the factorization and infeasibly hard if you do not.

Fractional and non-integer roots

Nothing stops the index from being fractional. The 2.5th root of x is x^(1/2.5) = x^0.4, and the 1.5th root of x is x^(2/3). Engineering correlations use these all the time: the Darcy friction factor for turbulent flow, heat transfer Nusselt number correlations, and sensor response curves often contain exponents like 0.8 or 1.25. If you need the 2.5th root of 32, the answer is 32^0.4 = 4, because 4^2.5 = 4² × √4 = 16 × 2 = 32.

Common traps to avoid

The biggest trap is distributing a root over a sum. √(a + b) is not equal to √a + √b. Check it: √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. Those are different numbers. Products are fine: √(a × b) = √a × √b for non-negative a and b, so √72 = √(36 × 2) = 6√2 is valid. Quotients are also fine: √(a ÷ b) = √a ÷ √b. The second trap is forgetting that every positive number has two real square roots. The principal root √x is always non-negative, but the equation x² = 25 has two solutions, x = 5 and x = −5. Algebra problems need both; a single √ symbol returns only one.

A note of history: YBC 7289

Babylonian scribes were computing square roots with impressive accuracy around 1700 BC. A small clay tablet in the Yale Babylonian Collection, catalogued as YBC 7289, shows a square with its diagonal labeled in sexagesimal (base 60) numerals. Decoded, the value is 1;24,51,10, which equals 1 + 24÷60 + 51÷3600 + 10÷216000 = 1.41421296. The true value of √2 is 1.41421356, so the tablet is correct to about six decimal places. The method they used is mathematically equivalent to one iteration of Newton's method starting from a reasonable guess, about 3,400 years before Newton wrote it down.

Root comparison across common inputs