Root Calculator

Calculate square root, cube root, nth root, and simplify radicals with our free root calculator

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What is a Root?

A root of a number is a value that, when raised to a certain power, gives the original number. The most common roots are square roots (2nd root) and cube roots (3rd root). Roots are the inverse operation of exponentiation.

Types of Roots

Square root (√) finds the number that, when squared, equals the original number. Cube root (³√) finds the number that, when cubed, equals the original number. Nth root finds the number that, when raised to the nth power, equals the original number.

Root Formula

The nth root of a number x is:

n√(x) = x^(1 ÷ n)

Examples of Root Calculations

Here are several everyday situations that show how easy it is to use our root calculator.

Example 1: Square Root

Calculate the square root of 25.

You can use our calculator by entering:

What is √25?

The calculator returns: 5

√25 = 5
5 × 5 = 25

The formula used is:

√(25) = 5

Example 2: Cube Root

Calculate the cube root of 27.

You can use our calculator by entering:

What is ³√27?

The calculator returns: 3

³√27 = 3
3 × 3 × 3 = 27

The formula used is:

3√(27) = 3

Example 3: Fourth Root

Calculate the 4th root of 16.

You can use our calculator by entering:

What is ⁴√16?

The calculator returns: 2

⁴√16 = 2
2⁴ = 16

The formula used is:

4√(16) = 2

Example 4: Simplify Radical

Simplify √72 to its simplest radical form.

You can use our calculator by entering:

What is √72 simplified?

The calculator returns: 6√2

Factor 72 = 36 × 2
√72 = √(36 × 2)
√72 = 6√2

The formula used is:

√(72) = 6√(2)

Example 5: UK Building Area

A square room in a London building has an area of 144 square metres. Find the side length.

You can use our calculator by entering:

What is the side length of a square with area 144 m²?

The calculator returns: 12 m

Area = side²
144 = side²
side = √144 = 12 m

The formula used is:

√(144) = 12

Example 6: UK Storage Volume

A cube-shaped storage unit in Manchester has a volume of 64 cubic metres. Find the edge length.

You can use our calculator by entering:

What is the edge length of a cube with volume 64 m³?

The calculator returns: 4 m

Volume = edge³
64 = edge³
edge = ³√64 = 4 m

The formula used is:

3√(64) = 4

Example 7: Distance Calculation

Calculate the distance using the Pythagorean theorem: √(3² + 4²).

You can use our calculator by entering:

What is √(9 + 16)?

The calculator returns: 5

√(9 + 16) = √25
√25 = 5

The formula used is:

√(9 + 16) = √(25) = 5

Example 8: Simplify √50

Simplify the square root of 50.

You can use our calculator by entering:

What is √50 simplified?

The calculator returns: 5√2

Factor 50 = 25 × 2
√50 = √(25 × 2)
√50 = 5√2

The formula used is:

√(50) = 5√(2)

Example 9: Fifth Root

Calculate the 5th root of 32.

You can use our calculator by entering:

What is ⁵√32?

The calculator returns: 2

⁵√32 = 2
2⁵ = 32

The formula used is:

5√(32) = 2

Example 10: Simplify √200

Simplify the square root of 200.

You can use our calculator by entering:

What is √200 simplified?

The calculator returns: 10√2

Factor 200 = 100 × 2
√200 = √(100 × 2)
√200 = 10√2

The formula used is:

√(200) = 10√(2)

Summary

These examples show how our root calculator can handle:

  • •Square root calculations (√25 = 5)
  • •Cube root calculations (³√27 = 3)
  • •Nth root calculations (⁴√16 = 2)
  • •Radical simplification (√72 = 6√2)

You can adapt any real-life situation involving roots to one of these patterns and let the calculator do the work for you.

How to Use

  1. Select the type of root calculation you need (square root, cube root, nth root, or simplify radical).
  2. Enter the number and any required parameters (e.g., root degree for nth root).
  3. View the result and step-by-step calculation breakdown.

Frequently asked questions

What is a square root?

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 × 5 = 25.

How do I calculate the nth root?

The nth root of a number is a value that, when raised to the power of n, gives the original number. For example, the 4th root of 16 is 2 because 2⁴ = 16.

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About this tool

Published by Online Tools Center

Last updated:

Check the units, assumptions and rounding before using a result. Health and financial estimates are educational and do not replace individual professional advice.

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