Nth Root Calculator – Find Any Root Value Instantly

Find the nth root of any number in seconds with our Nth Root Calculator. Whether you’re a student learning algebra, an engineer performing scientific calculations, or a mathematician dealing with complex expressions, this tool helps you calculate roots accurately, instantly, and effortlessly.

Our calculator supports any real number and any positive integer as a root — including square roots, cube roots, fourth roots, and beyond.

What Is an Nth Root?

The nth root of a number is a value that, when raised to the power of n, equals the original number.

Mathematically, the nth root of a number x is written as:

√ₙx = x^(1/n)

That means:
If yⁿ = x, then y = √ₙx

Example:

The 3rd root (cube root) of 8 is 2, because 2³ = 8.
Therefore, √₃8 = 2.

This concept is fundamental in algebra, geometry, physics, and many scientific applications where you need to reverse an exponent or find a proportional rate.

Nth Root Calculator — Simple, Fast, and Precise

Our Nth Root Calculator instantly computes any root of any number — even decimals, negatives (for odd roots), and fractions.

How to Use:

1. Enter the number (x) you want to find the root of.
2. Enter the degree (n) — e.g., 2 for square root, 3 for cube root, etc.
3. Click Calculate.
4. Instantly get your nth root value with step-by-step explanations.

Example Inputs

Nth Root Formula

The nth root of a number x is calculated using the formula:

√ₙx = x^(1/n)

This is equivalent to raising x to the reciprocal of n.

Example 1:

Find the 5th root of 32.
√₅32 = 32^(1/5) = 2

Since 2⁵ = 32, the answer is 2.

Example 2:

Find the 4th root of 81.
√₄81 = 81^(1/4) = 3

Since 3⁴ = 81, the answer is 3.

Different Types of Roots

The Nth Root Calculator supports all root types — from the common square and cube roots to higher-order and fractional roots.

1. Square Root (2nd Root)

The square root of a number x is a value that, when squared, equals x.

Formula:
√x = x^(1/2)

Example:
√49 = 7 because 7² = 49

Used in: Geometry, area calculations, statistics (standard deviation), and quadratic equations.

2. Cube Root (3rd Root)

The cube root of x is the number that, when cubed, equals x.

Formula:
³√x = x^(1/3)

Example:
³√27 = 3 because 3³ = 27

Used in: Volume calculations, scaling laws, and physics.

3. Fourth Root and Higher Roots

These represent less common but mathematically significant roots used in advanced science and engineering.

Formula:
√ₙx = x^(1/n)

Examples:
√₄16 = 2
√₅32 = 2

Used in: Signal processing, energy functions, and complex modeling.

Negative Numbers and Roots

Handling negative numbers depends on whether the root index is even or odd:

In other words:

• Even roots of negative numbers are imaginary (complex numbers).
• Odd roots of negative numbers are real.

Step-by-Step Example Calculations

Example 1: Square Root

Find √144.

Formula:
√144 = 144^(1/2) = 12

Result: 12

Example 2: Cube Root

Find³√125.

Formula:
³√125 = 125^(1/3) = 5

Result: 5

Example 3: Fourth Root

Find √₄81.

Formula:
√₄81 = 81^(1/4) = 3

Result: 3

Example 4: Odd Root of a Negative Number

Find ³√(-8).

Formula:
³√(-8) = -8^(1/3) = -2

Result: -2

Example 5: Decimal Example

Find ³√0.008.

Formula:
0.008^(1/3) = 0.2

Result: 0.2

Fractional Roots

The nth root of a fraction can be computed by taking the root of both numerator and denominator separately.

Formula:
√ₙ(a/b) = (√ₙa) / (√ₙb)

Example:
√₃(8/27) = √₃(8) / √₃(27) = 2/3

Result: 2/3

Relationship Between Exponents and Roots

Roots are the inverse operation of exponents.

If:
xⁿ = a
then
√ₙa = x

That means:

• Raising reverses a root.
• Rooting reverses a power.

Example:
(√₃8)³ = 8
(8³)^(1/3) = 8

Understanding this relationship helps in solving exponential and radical equations efficiently.

Applications of Nth Roots in Real Life

Nth roots are fundamental in many real-world fields beyond pure mathematics.

1. Mathematics

Simplifies algebraic equations and radicals; used in polynomial roots and logarithmic conversions.

2. Geometry

Used to calculate side lengths, areas, and volumes, especially when reverse-engineering dimensions.

3. Physics

Applied in formulas involving energy, acceleration, and proportional scaling.

4. Engineering

Nth roots help compute mechanical stress, resistance, and efficiency in design systems.

5. Statistics

Standard deviation and variance calculations often involve square roots.

6. Finance

Used in compound interest and geometric mean calculations:

Formula for annual growth rate:
r = (FV / PV)^(1/n) - 1

Key Properties of Roots

These rules simplify complex radical expressions and make manual calculations easier.

How to Use the Nth Root Calculator

Our Nth Root Calculator is designed to deliver precision and ease of use.

Steps:

1. Enter the number (x).
2. Enter the root degree (n).
3. Click Calculate.
4. Instantly get your nth root value.
5. Optionally view the step-by-step calculation for learning.

You can use it for integers, decimals, fractions, and even negative numbers (for odd roots).

Why Use the Digital Calculator Nth Root Calculator?

We built our Nth Root Calculator to combine simplicity, accuracy, and learning support in one place.

Key Features:

• Handles any degree of root (square, cube, fourth, fifth, etc.).
• Accepts negative and fractional inputs.
• Delivers instant results with clear explanations.
• Educational interface ideal for students and professionals.
• Mobile-friendly — works seamlessly across devices.
• 100% free, no sign-up required.

Use it to solve equations, check homework, or perform quick technical computations with confidence.Try the Nth Root Calculator now and see how easy it is to find roots of any number.

Comparison: Common Root Values

Working with Imaginary Roots

For even roots of negative numbers, results are imaginary.
The imaginary unit i represents √(-1).

Examples:

• √(-9) = 3i
• √₄(-16) = 2i

Our Nth Root Calculator supports real numbers. For complex roots, you can use our Complex Number Calculator..

Solving Nth Root Equations

Nth roots often appear in algebraic equations. You can isolate variables using inverse exponent rules.

Example:
Solve for x:
x³ = 125
Take cube root both sides:
x = √₃125 = 5

For fractional powers:
x^(1/4) = 3
Raise both sides to 4:
x = 3⁴ = 81

Understanding nth roots simplifies equations of exponential and radical types across science and engineering.