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Algebra & RadicalsHigh-Precision Simplifier

Root Calculator

Calculate square roots (√x), cube roots (∛x), and general nth roots (ⁿ√x) with exact radical simplification, prime factorization, and decimal approximations.

√

Square Root Calculator

Find principal and simplified square root of any real number: y = √x = x^(1/2)

Index n = 2
Presets:
∛

Cube Root Calculator

Find exact simplified cube roots of positive and negative numbers: y = ∛x = x^(1/3)

Index n = 3
3
Presets:
ⁿ√

General Root Calculator (nth Root)

Compute any arbitrary degree root: y = ⁿ√x = x^(1/n)

Arbitrary Index n
Presets:
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Understanding Roots and Radicals

In mathematics, the nth root of a number x is a value y which, when raised to the power of n, equals x:

yn = x ⇔ y = n√x = x1/n
  • Radical Symbol (√): The mathematical symbol used to denote root extraction.
  • Index / Degree (n): The superscript integer indicating which root is being computed. If no index is shown (e.g. √x), it implicitly represents the square root (n = 2).
  • Radicand (x): The value positioned inside the radical symbol.
  • Vinculum: The horizontal overbar extending across the top of the radicand.

Essential Properties & Rules of Radicals

Product Rule of Radicals
ⁿ√(a · b) = ⁿ√a · ⁿ√b

The root of a product equals the product of the individual roots.

Quotient Rule of Radicals
ⁿ√(a / b) = (ⁿ√a) / (ⁿ√b)

The root of a quotient equals the quotient of the numerator and denominator roots.

Power of a Root
(ⁿ√a)ᵐ = ⁿ√(aᵐ) = a^(m/n)

Roots convert directly to rational fractional exponents.

Nested / Repeated Roots
ᵐ√(ⁿ√a) = ᵐⁿ√a = a^(1/(m·n))

Nested radicals multiply their indices together.

Common Roots Reference Tables

Perfect Squares (1 to 20)

Radicand (x)Square Root (√x)Radicand (x)Square Root (√x)
1112111
4214412
9316913
16419614
25522515
36625616
49728917
64832418
81936119
1001040020

Perfect Cubes (1 to 10)

Radicand (x)Cube Root (∛x)Negative (∛-x)
11-1
82-2
273-3
644-4
1255-5
2166-6
3437-7
5128-8
7299-9
1,00010-10

How to Simplify Radicals Step-by-Step

To simplify a radical of the form n√x into its simplest radical form k n√r:

  1. Find the Prime Factorization of the Radicand: Break x into prime factors (e.g. 72 = 2³ × 3²).
  2. Group Factors into Powers of the Index (n): For square root (n = 2), group into pairs of identical factors: 72 = (2² × 3²) × 2 = (6²) × 2.
  3. Extract Perfect nth Powers Outside the Radical: Move the base of each complete power outside the radical sign: √72 = √(6² × 2) = 6√2.
  4. Multiply Remaining Factors Inside: Keep factors with exponents less than n inside the radical.

Frequently Asked Questions

What is the principal square root?

The principal square root is the non-negative square root of a non-negative real number. While the equation y² = 25 has two solutions (y = +5 and y = -5), the radical expression √25 uniquely denotes the principal positive value +5.

Can you take the square root of a negative number?

In the real number system (ℝ), no real number multiplied by itself gives a negative result. However, in the complex number system (ℂ), the square root of a negative number is an imaginary number using i = √(-1) (e.g. √(-25) = 5i).

Why can cube roots have negative real answers?

Because an odd number of negative factors produces a negative product: (-3) × (-3) × (-3) = -27. Therefore, the cube root of -27 is a real number equal to -3.

What happens if the root degree is zero (n = 0)?

The 0th root of a number (⁰√x = x^(1/0)) is mathematically undefined because it involves division by zero in the exponent.