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n-th Root Calculator

Calculate the n-th root of any number with high precision up to 1000 decimal places. Features step-by-step solutions, visual diagrams, and mathematical notation in both radical and exponential forms.

Free to useNo sign-up requiredUpdated Jan 2026
n-th Root CalculatorTry it now — free ▼
Quick Examples
∛8 =2 ∜16 =2 √100 =10 ∛-27 =-3 √2 ≈1.414 ⁶√10⁶ =10

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About n-th Root Calculator

Welcome to the n-th Root Calculator, a professional-grade mathematical tool for computing roots of any index with exceptional precision. Whether you need a simple square root or a high-precision 100th root, this calculator uses advanced algorithms to deliver accurate results up to 1000 decimal places.

What is an n-th Root?

The n-th root of a number x (written as n√x or x1/n) is a value r that, when raised to the power n, equals x. In mathematical notation:

n-th Root Definition
$$\sqrt[n]{x} = r \quad \text{where} \quad r^n = x$$

Common roots you may encounter include:

Radical Form vs Exponential Form

There are two equivalent ways to express n-th roots:

Equivalent Notations
$$\sqrt[n]{x} = x^{\frac{1}{n}}$$

Radical form (√) is intuitive and commonly used in basic mathematics. Exponential form (x1/n) follows the rules of exponents and is preferred in algebra and calculus because it integrates seamlessly with other exponential operations.

Properties of n-th Roots

Handling Negative Numbers

The behavior of roots depends on whether the index n is odd or even:

Odd roots (n = 1, 3, 5, 7...): Every real number has exactly one real n-th root. The cube root of -8 is -2 because (-2)³ = -8.

Even roots (n = 2, 4, 6, 8...): Negative numbers have no real even roots. The square root of -4 does not exist in real numbers. In complex numbers, √(-4) = 2i, where i is the imaginary unit (i² = -1).

How This Calculator Works

  1. Enter the radicand: Input the number you want to find the root of. Accepts positive numbers, negative numbers (for odd roots), decimals, and scientific notation.
  2. Specify the root index: Enter the value of n. Use 2 for square root, 3 for cube root, etc.
  3. Select precision: Choose from 10 to 1000 decimal places depending on your accuracy requirements.
  4. Calculate: The tool uses the Newton-Raphson iterative algorithm with arbitrary-precision decimal arithmetic to compute your result.

The Newton-Raphson Algorithm

This calculator employs the Newton-Raphson method (also known as Newton's method) to iteratively converge on the precise root value:

Newton-Raphson Iteration
$$y_{k+1} = \frac{(n-1) \cdot y_k + \frac{x}{y_k^{n-1}}}{n}$$

Starting from an initial estimate, each iteration refines the answer until the desired precision is achieved. This method converges rapidly—typically requiring only a few dozen iterations even for 1000-digit precision.

Perfect n-th Powers

A perfect n-th power is an integer whose n-th root is also an integer. Recognizing these can simplify calculations:

Applications of n-th Roots

Frequently Asked Questions

What is an n-th root?

The n-th root of a number x is a value r such that r raised to the power n equals x. Written mathematically as ⁿ√x = r, where rⁿ = x. For example, the cube root (3rd root) of 8 is 2 because 2³ = 8. Common roots include square root (n=2), cube root (n=3), and fourth root (n=4).

Can you take an even root of a negative number?

In real numbers, you cannot take an even root (like square root, 4th root) of a negative number because no real number multiplied by itself an even number of times produces a negative result. However, in complex numbers, even roots of negatives exist as imaginary numbers. For example, √(-1) = i, where i is the imaginary unit.

What is the difference between radical and exponential form?

Radical form uses the root symbol (√) to express roots, like ³√27. Exponential form uses fractional exponents, like 27^(1/3). Both represent the same mathematical operation: ⁿ√x = x^(1/n). Exponential form is often preferred in advanced mathematics because it follows the standard rules of exponents.

How is n-th root calculated with high precision?

High-precision n-th roots are calculated using iterative algorithms like Newton-Raphson (also called Newton's method). Starting with an initial estimate, the algorithm repeatedly refines the answer using the formula: y_(k+1) = ((n-1)·y_k + x/y_k^(n-1))/n. This converges quickly to the true root with arbitrary precision using decimal arithmetic libraries.

What are perfect n-th powers?

A perfect n-th power is an integer that can be expressed as another integer raised to the n-th power. For example: perfect squares (4, 9, 16, 25...), perfect cubes (8, 27, 64, 125...), and perfect fourth powers (16, 81, 256...). The n-th root of a perfect n-th power is always an integer.

Additional Resources

Reference this content, page, or tool as:

"n-th Root Calculator" at https://MiniWebtool.com/nth-root-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 27, 2026

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

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