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Home Page > Math > Basic Math Operations

n-th Root Calculator

Enter a number and an index n. Get the n-th root in radical and exponential form, with as many as 1000 decimal places, and the steps.

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The n-th root is the factor used n times: ⁴√81 = 3 = 81^(1/4)

Enter a number and an index n to get the n-th root in radical and exponential form, with up to 1000 decimal places and the steps.

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

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