Simplifying Nth Roots with Variables Calculator

What is an nth root?

An nth root of a number is a value that, when raised to the nth power, gives the original number. For example, the cube root of 8 is 2 because 2³ = 8.

When working with variables, we often need to simplify expressions like √(x³) or ³√(x⁶). The process involves factoring the radicand (the number inside the root) and separating it into perfect powers.

Simplifying radicals with variables

To simplify √(x³), follow these steps:

1. Factor the radicand: x³ = x² × x
2. Separate the square root: √(x² × x) = √(x²) × √x
3. Simplify: √(x²) = x, so the simplified form is x√x

For cube roots (³√), the process is similar but involves perfect cubes:

1. Factor the radicand: x⁶ = x³ × x³
2. Separate the cube root: ³√(x³ × x³) = ³√(x³) × ³√(x³)
3. Simplify: ³√(x³) = x, so the simplified form is x²

Using the calculator

Our calculator simplifies nth roots with variables by following the same steps as shown above. Simply enter the radicand and the root index, and the calculator will show you the simplified form.

Examples

Example 1: Square root of x³

Input: √(x³)

Calculation: √(x² × x) = x√x

Result: x√x

Example 2: Cube root of x⁶

Input: ³√(x⁶)

Calculation: ³√(x³ × x³) = x²

Result: x²

Example 3: Fourth root of x⁸

Input: ⁴√(x⁸)

Calculation: ⁴√(x⁴ × x⁴) = x²

Result: x²