Find any root (4th root, 5th root, 6th root, etc.) of numbers with precision. Compute the nth root using the formula x^(1/n) and verify your result by raising it back to the power N.
Example 1: 4th root of 81
Formula: 811/4
Solution: Since 3⁴ = 3 × 3 × 3 × 3 = 81, the 4th root of 81 is 3.
Answer: ⁴√81 = 3
Example 2: 5th root of 32
Formula: 321/5
Solution: Since 2⁵ = 2 × 2 × 2 × 2 × 2 = 32, the 5th root of 32 is 2.
Answer: ⁵√32 = 2
Example 3: 3rd root (cube root) of 125
Formula: 1251/3
Solution: Since 5³ = 5 × 5 × 5 = 125, the cube root of 125 is 5.
Answer: ³√125 = 5
Example 4: 6th root of 64
Formula: 641/6
Solution: Since 2⁶ = 2 × 2 × 2 × 2 × 2 × 2 = 64, the 6th root of 64 is 2.
Answer: ⁶√64 = 2
Example 5: 4th root of 10000
Formula: 100001/4
Solution: Since 10⁴ = 10 × 10 × 10 × 10 = 10000, the 4th root of 10000 is 10.
Answer: ⁴√10000 = 10
The nth root of a number x is a value r such that r raised to the power n equals x. In other words, r is the number that, when multiplied by itself n times, gives x.
The nth root can be expressed as raising the number x to the power of 1/n. This is the most commonly used form for computation.
The nth root of a product equals the product of the nth roots. This property is useful for simplifying radical expressions.
Raising the nth root of x to the power m is equivalent to computing x raised to the power m/n.
To verify the result, raise the computed root back to the power N. The result should equal the original number x (within floating-point precision).
If a number x is a perfect power (e.g., 32 = 2⁵), the nth root is an integer. Memorizing common powers of small integers helps estimate roots quickly.
For non-perfect powers, the nth root is typically an irrational number. You can often simplify by factoring: ⁴√32 = ⁴√(16 × 2) = 2 × ⁴√2
To estimate ⁿ√x, find nearby perfect powers. For example, ⁴√100 is between ⁴√81 (3) and ⁴√256 (4), so it's approximately 3.2.
Our nth root calculator provides a comprehensive set of features to help you compute and understand roots of any index:
Get nth root results instantly using JavaScript's Math.pow for fast, accurate computation.
Every result is verified by raising it back to the power N, confirming the calculation is correct.
Calculate any root — square root (2nd), cube root (3rd), 4th root, 5th root, and beyond.
Choose decimal precision from 2 to 12 decimal places for precise calculations.
View the detailed calculation steps showing the formula and verification process.
No registration required. Use the nth root calculator as many times as you need, completely free.
An nth root of a number x is a value r that satisfies the equation rn = x. In simpler terms, when you multiply r by itself n times, you get x. The nth root is denoted using the radical symbol with an index: n√x, or using fractional exponents: x1/n.
For example, the 4th root of 81 is 3, because 3 × 3 × 3 × 3 = 81. The 5th root of 32 is 2, because 2 × 2 × 2 × 2 × 2 = 32. The nth root generalizes the concept of square roots (where n = 2) and cube roots (where n = 3) to any positive integer index.
Nth roots are fundamental in many areas of mathematics and its applications, including algebra, geometry, physics, engineering, finance, and data science. Understanding nth roots is essential for working with polynomial equations, growth models, fractals, signal processing, and many other advanced topics.
Using our nth root calculator is straightforward. Follow these simple steps:
Type or paste a positive number into the first input field. You can enter integers (like 81), decimals (like 16.5), or any non-negative real number.
Specify which root you want to calculate. For a 4th root, enter 4. For a 5th root, enter 5. The index must be a positive integer (1, 2, 3, 4, ...).
Choose how many decimal places you'd like in your result — from 2 to 12 decimal places. The default is 6 decimal places.
Click the "Calculate Nth Root" button or press Enter. Your result will appear instantly, along with verification showing that (result)N equals the original number.
Here is a reference table of common perfect powers and their nth roots:
| Number (x) | Root Index (N) | Nth Root | Verification |
|---|---|---|---|
| 16 | 2 (Square Root) | 4 | 4² = 16 |
| 27 | 3 (Cube Root) | 3 | 3³ = 27 |
| 81 | 4 (4th Root) | 3 | 3⁴ = 81 |
| 32 | 5 (5th Root) | 2 | 2⁵ = 32 |
| 64 | 6 (6th Root) | 2 | 2⁶ = 64 |
| 128 | 7 (7th Root) | 2 | 2⁷ = 128 |
| 256 | 8 (8th Root) | 2 | 2⁸ = 256 |
| 512 | 9 (9th Root) | 2 | 2⁹ = 512 |
| 1024 | 10 (10th Root) | 2 | 2¹⁰ = 1024 |
| 10000 | 4 (4th Root) | 10 | 10⁴ = 10000 |
Nth roots have numerous practical applications across various fields:
Important: This nth root calculator is designed for positive real numbers with positive integer root indices. Nth roots of negative numbers with even indices involve complex numbers and are not computed by this tool.
This nth root calculator was designed to provide accurate, educational root calculations for students, teachers, engineers, and anyone needing to compute roots of any index. Our tool combines modern JavaScript computation with clear step-by-step verification to deliver both speed and educational value.
Disclaimer: While we strive to make our nth root calculator as accurate and reliable as possible, this tool is for educational and general use purposes. For critical scientific, engineering, or financial calculations, please verify results independently or consult with appropriate professionals.