Root Calculator
Calculate any root (square root, cube root, nth root) with step-by-step solutions. Find roots of numbers instantly.
ⁿ√x = x^(1/n)Root Result
3√27
3
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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
?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
Root Result
3√27
3