Take The Nth Root Calculator

The nth root calculator helps you find the root of a number when you know the index. Whether you're solving math problems, analyzing data, or working with scientific calculations, this tool provides quick and accurate results.

What is 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 cube root of 27 is 3 because 3 × 3 × 3 = 27. The nth root is a fundamental concept in mathematics with applications in algebra, calculus, and many other fields.

There are special names for specific roots:

Square root (2nd root)
Cube root (3rd root)
Fourth root (4th root)
Fifth root (5th root)

For roots with even indices, the result is always positive. For roots with odd indices, the result can be positive or negative depending on the original number.

How to calculate the nth root

Calculating the nth root manually can be complex, especially for higher indices. However, with the right tools and understanding, you can find roots accurately. Here's a step-by-step guide:

Identify the number you want to find the root of (the radicand).
Determine the index (n) of the root you want to find.
Use the nth root formula or a calculator to find the result.
Verify your result by raising it to the power of n to ensure it equals the original number.

For example, to find the 3rd root of 64:

The radicand is 64.
The index is 3.
Using the formula, the result is 4 because 4 × 4 × 4 = 64.

Formula for the nth root

Nth Root Formula

The nth root of a number x can be calculated using the following formula:

y = x^(1/n)

Where:

y = the nth root of x
x = the radicand (the number under the root)
n = the index of the root

This formula is derived from the definition of roots and exponents. It allows you to find roots of any index using basic exponentiation.

Examples of nth roots

Here are some examples of nth roots calculated using the formula:

Radicand (x)	Index (n)	Nth Root (y)	Verification
16	2	4	4 × 4 = 16
81	2	9	9 × 9 = 81
27	3	3	3 × 3 × 3 = 27
125	3	5	5 × 5 × 5 = 125
16	4	2	2 × 2 × 2 × 2 = 16

These examples demonstrate how the nth root formula works in practice. You can use this calculator to find roots for any number and index.

FAQ

What is the difference between a square root and a cube root?

The square root is the 2nd root of a number, while the cube root is the 3rd root. The square root of a number x is a value y such that y × y = x. The cube root of a number x is a value y such that y × y × y = x.

Can the nth root of a negative number be real?

Yes, the nth root of a negative number can be real if the index n is odd. For example, the cube root of -8 is -2 because (-2) × (-2) × (-2) = -8. However, roots with even indices are always positive.

How do I calculate the nth root of a number using a calculator?

Most scientific calculators have a root function that allows you to find the nth root of a number. Simply enter the number, press the root button, and then enter the index. Alternatively, you can use the exponentiation function with the formula y = x^(1/n).