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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
√=2
√=3
√=4
√=5
√=6
√=7
√=8
√=9
√=10
Perfect Cubes
³√=1
³√=2
³√=3
³√=4
³√=5
³√=6
³√=7
³√=8
³√=9
³√=10
Root Result
3√27
3
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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
What is a root in mathematics?
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.
What is the formula for nth root?
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.
Can you take even roots of negative numbers?
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.
What is a perfect root?
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.
How do I simplify a radical?
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: 2026-01-15
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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
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