Root Calculator
Calculate any root (square root, cube root, nth root) with step-by-step solutions. Find roots of numbers instantly.
Formula:ⁿ√x = x^(1/n)
Root Result
3√27
3
TypePerfect root (integer)
Verification: 3^3 = 27.000000
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Step-by-Step Solution
Step 1: Apply nth root formula
3√27 = 27^(1/3)
Step 2: Convert to decimal exponent
1/3 = 0.333333
Step 3: Calculate
27^0.333333 = 3
Result: 3√27 = 3
Perfect Roots Reference
Perfect Squares
1
√=1
4
√=2
9
√=3
16
√=4
25
√=5
36
√=6
49
√=7
64
√=8
81
√=9
100
√=10
Perfect Cubes
1
³√=1
8
³√=2
27
³√=3
64
³√=4
125
³√=5
216
³√=6
343
³√=7
512
³√=8
729
³√=9
1000
³√=10
Root Result
3√27
3
TypePerfect root (integer)
Verification: 3^3 = 27.000000
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?How Do You Calculate Roots?
The nth root of a number x is a value that, when raised to the power n, equals x. Square root (n=2): sqrt(9) = 3 because 3^2 = 9. Cube root (n=3): cbrt(8) = 2 because 2^3 = 8. Formula: nth root of x = x^(1/n). Perfect squares have integer square roots; others are irrational.
What is a Root?
A root is the inverse operation of raising a number to a power. The nth root of x (written as x^(1/n) or with a radical symbol) is the number that, when multiplied by itself n times, equals x. Roots are fundamental in algebra, geometry (calculating distances and areas), and solving polynomial equations.
Key Facts About Roots
- Square root: sqrt(x) = x^(1/2). sqrt(16) = 4 because 4^2 = 16
- Cube root: cbrt(x) = x^(1/3). cbrt(27) = 3 because 3^3 = 27
- Nth root: x^(1/n) is the number that when raised to n gives x
- Perfect squares (1, 4, 9, 16, 25...) have integer square roots
- Perfect cubes (1, 8, 27, 64, 125...) have integer cube roots
- Square root of negative numbers requires complex numbers (imaginary unit i)
- Cube roots of negative numbers are negative real numbers
- Simplify radicals by factoring out perfect powers: sqrt(50) = 5*sqrt(2)
Quick Answer
The nth root of a number x is a value that, when raised to the power n, equals x. Square root (n=2): sqrt(9) = 3 because 3^2 = 9. Cube root (n=3): cbrt(8) = 2 because 2^3 = 8. Formula: nth root of x = x^(1/n). Perfect squares have integer square roots; others are irrational.
Frequently Asked Questions
A root is the inverse of an exponent. The nth root of a number x is a value that, when raised to the power n, gives x. For example, the square root (2nd root) of 9 is 3, because 3² = 9. The cube root (3rd root) of 8 is 2, because 2³ = 8.
The nth root of x equals x raised to the power 1/n. Mathematically: ⁿ√x = x^(1/n). For example, the cube root of 27 = 27^(1/3) = 3. This formula works for any positive root.
No, even roots (square root, 4th root, etc.) of negative numbers don't exist in real numbers - they result in complex/imaginary numbers. However, odd roots (cube root, 5th root, etc.) of negative numbers do exist. For example, ³√(-8) = -2.
A perfect root occurs when the nth root of a number is a whole number. Perfect squares: 1, 4, 9, 16, 25... Perfect cubes: 1, 8, 27, 64, 125... These are numbers whose roots are integers.
To simplify √n, find the largest perfect square factor. For example, √72 = √(36×2) = √36 × √2 = 6√2. For cube roots, find perfect cube factors. √50 = √(25×2) = 5√2.
Last updated: 2025-01-15
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How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
Joseph OrdunaFounder & Software Engineer
Full-stack software engineer specializing in embedded systems, web architecture, and AI/ML. Founder of Practical Web Tools. Built the gesture-controlled drone IP acquired by KD Interactive (Aura Drone, sold on Amazon).
Root Result
3√27
3
TypePerfect root (integer)
Verification: 3^3 = 27.000000