Real Nth Roots Calculator

The real nth root of a number is a value that, when raised to the nth power, gives the original number. This calculator finds real roots for positive real numbers and positive integer exponents.

What is an nth root?

The nth root of a number x is a value y such that y^n = x. For example, the cube root of 27 is 3 because 3^3 = 27.

For even roots (like square roots), the result is always non-negative. For odd roots, the sign of the original number determines the sign of the root.

How to calculate nth roots

To find the nth root of a number x:

Verify that x is a positive real number and n is a positive integer
Use the formula: y = x^(1/n)
For even roots, take the positive result

Formula

The nth root of x is calculated as:

y = x^(1/n)

Assumptions

x must be a positive real number
n must be a positive integer
For even roots, only the positive root is returned

Real vs. complex roots

Real nth roots exist for all positive real numbers when n is a positive integer. For example:

The square root of 4 is 2 (real)
The cube root of -8 is -2 (real)
The fourth root of 16 has two real roots: 2 and -2

Complex roots exist for negative numbers when n is even, but this calculator only returns real roots.

Practical applications

Nth roots are used in various mathematical and practical contexts:

Finding dimensions in geometry
Calculating average rates in physics
Solving polynomial equations
Working with exponential growth and decay

FAQ

What is the difference between square root and cube root?

The square root (2nd root) of a number x is a value y such that y² = x. The cube root (3rd root) is a value y such that y³ = x. The square root always gives one non-negative result, while the cube root preserves the sign of the original number.

Can I find roots of negative numbers?

This calculator only finds real roots. For negative numbers with even exponents, you'll need to use complex numbers. For odd exponents, the result will be negative.

What happens when the exponent is 1?

When the exponent is 1, the "1st root" is simply the number itself, as x^(1/1) = x.