Radical & Root Calculator

Solve for the square root, cube root, or any custom nth-degree radical of a positive or negative number.

Preparing Root Calculator

Please wait ...

Input

Enter the base value and the root degree (like 2 for square, 3 for cube, or 4 for fourth root) below to determine the root value.

Input summary

Your calculator summary shows here.

Step by Step Calculation

ⁿ√x

Step by step calculation shows here

Calculate first and the full working will appear below automatically.

A Guide to Roots and Radical Math

In algebra, finding the root of a number is the direct reverse operation of raising a number to an exponent power. When you square a number (like multiplying 4 by itself to get 16), the inverse operation is finding the square root (which asks: "what number multiplied by itself equals 16?"). Roots are fundamental in geometry formulas, physical dynamics, statistical deviations, and complex financial compound calculations.

How Roots and Exponents Relate

A root is written using a radical symbol (√) with a small number called the index or degree indicating the depth of the root. If no index is written, it is assumed to be a square root (degree 2). A cube root has a degree of 3, a fourth root has a degree of 4, and so on.

Mathematically, taking the nth root of a number is exactly the same as raising that number to a fractional exponent. For example, the square root of a number is the same as raising it to the power of 1/2, and the cube root is raising it to the power of 1/3. To explore power calculations directly, check out our raising numbers to powers solver.

If you are solving perfect square factors manually, it helps to break down the number, which you can resolve using our dividing numbers into prime components.

Everyday Applications of Roots

Sizing Flat Plots: If you know the square footage of a square piece of land, finding its boundary side lengths requires a square root. You can check sizes using our flat shapes area solver.
Diagonal Cut Calculations: Solving diagonal bracing lengths in construction uses roots. You can find these lengths using our hypotenuse calculation solver.
Financial Growth Rates: Compound annual growth rates require roots to calculate average returns over multiple years, which you can double-check with our standard daily math tools.
Data Spread Variations: Statistical measures of dispersion (like standard deviation) use square roots to return variations back to original units.

Rational vs. Irrational Roots

Some roots result in clean whole numbers, known as perfect roots. For example, the square root of 9 is 3, and the cube root of 8 is 2, because these bases are formed by squaring or cubing whole numbers.

However, most numbers do not have perfect roots. The square root of 2 is approximately 1.41421, which is an irrational number that continues forever without repeating. Our online solver displays highly accurate decimal expansions for non-perfect roots.

Example of Sizing a Square Garden

Suppose a gardener has a square patch of land with a total area of 144 square feet and wants to buy fencing for one side of the garden.

Because the garden is a perfect square, the area equals the side length squared (Area = side²). To find the side length, we calculate the square root of 144. The calculator finds that the square root of 144 is exactly 12, because 12 × 12 = 144. The gardener needs exactly 12 feet of fencing for one side. This practical example shows how roots translate area capacities back into linear measurements.