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Calculator Answers

Radical Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A radical calculator simplifies mathematical roots of any positive integer index. It performs prime factorization on the radicand, groups terms based on the radical index to extract coefficients, and displays the exact simplified radical along with a 6-place decimal approximation.

Simplify radicals, square roots, cube roots, and higher-order nth roots to their simplest radical form. Features prime factorization step-by-step breakdowns and decimal approximations.

See also: Exponent Calculator, Square Root Calculator, and Net Ionic Equation Calculator.

Quick Answer

Simplify radicals, square roots, cube roots, and nth roots to their lowest radical form with step-by-step work.

Radical Inputs

Optional · e.g. 1

Negative values ok for odd index · e.g. 72

Square = 2, Cube = 3 · e.g. 2

Decimal Places
Radical Solution

Simplified Expression

6·√(2)

Decimal value ≈ 8.485281

Original Radical√(72)
Decimal Value8.485281
Is Perfect PowerNo
Simplified Coeff6
Remaining Radicand2

Radicals are numbers that, when raised to the power of the index, yield the radicand.

Step-by-Step Factoring

[1]Find the prime factorization of |N| (72):
[2] - 72 = 2^3 × 3^2
[3]Extract groups of size k (2) from each prime power:
[4] - Prime 2: exponent is 3. Extract 2^2 as 2^1 = 2 outside.
[5] - Prime 3: exponent is 2. Extract 3^2 as 3^1 = 3 outside.
[6]Apply initial coefficient and signs:
[7] - Simplified outside coefficient: 1 (initial) × 6 (extracted) = 6
[8] - Remainder inside the radical: 2
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Examples

Simplify a square root: √72

√72 = √(36 × 2) = 6√2 ≈ 8.485281

Simplify a cube root: ∛54

∛54 = ∛(27 × 2) = 3∛2 ≈ 3.779763

Simplify negative cube root: ∛-250

∛-250 = -∛(125 × 2) = -5∛2 ≈ -6.299605

How it works

Radical simplification uses prime factorization to group matching factors:

Radical Simplifier General Form

k√(ak · b) = a · k√b

Where k is the index, ak is a perfect power extracted, and b is the remaining factor.

How Radical Extraction Works

Every positive integer can be broken down into prime factors. For a radical of index k, we look at the exponent of each prime factor. If a prime p appears with an exponent c ≥ k, we can write it as p^c = p^(k · m + r). We can extract p^m outside the radical sign and leave p^r inside. Combining these products for all prime factors gives the simplified radical form.

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Square Root CalculatorCube Root CalculatorExponent Calculator

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Frequently asked questions

To simplify a radical, you factor the number under the radical (the radicand) to find perfect powers that match the index of the radical. For example, to simplify √72 (square root index 2), you find the largest perfect square factor, which is 36 (36 × 2 = 72). You take the square root of 36 (which is 6) and move it outside the radical, leaving the remaining factor inside: 6√2.

The index is the small number written outside the radical symbol (e.g., the '3' in ∛x). It indicates which root you are finding. If no index is written, it is assumed to be 2 (a square root). For index 'n', you are finding the number that must be multiplied by itself 'n' times to equal the radicand.

Yes, but only if the index of the radical is an odd number (such as 3 for cube root, 5 for fifth root, etc.). Odd roots of negative numbers yield negative real results (e.g., ∛-8 = -2). Even roots of negative numbers (like √-4) are not real numbers and result in complex or imaginary numbers.

A square root has an index of 2 and solves for y in the equation y² = x. A cube root has an index of 3 and solves for y in the equation y³ = x. For example, √9 = 3 (since 3 × 3 = 9) while ∛27 = 3 (since 3 × 3 × 3 = 27).