Simplify Radical Expressions Calculator Cube Root

This guide explains how to simplify radical expressions and calculate cube roots using our free online calculator. Whether you're studying algebra or need to simplify √x or ∛x, this tool will help you understand the process step by step.

How to Use This Calculator

Our simplify radical expressions calculator is designed to be user-friendly. Follow these steps to get accurate results:

Enter the number you want to simplify in the input field.
Select whether you want to simplify a square root (√x) or a cube root (∛x).
Click the "Calculate" button to see the simplified form.
Review the result and the step-by-step explanation.

The calculator will display the simplified radical expression and explain how it was derived. You can also use the reset button to clear the input and start over.

How to Simplify Radical Expressions

Simplifying radical expressions involves reducing the radical to its simplest form. Here's a step-by-step guide:

Step 1: Factor the Radicand

First, factor the number inside the radical (the radicand) into perfect squares (for square roots) or perfect cubes (for cube roots).

Example for Square Root

Simplify √72:

72 = 36 × 2 = 6² × 2

√72 = √(36 × 2) = √36 × √2 = 6√2

Step 2: Separate the Radical

Once you've factored the radicand, separate the perfect square (or cube) from the remaining factor.

Example for Cube Root

Simplify ∛108:

108 = 27 × 4 = 3³ × 4

∛108 = ∛(27 × 4) = ∛27 × ∛4 = 3∛4

Step 3: Simplify the Radical

Take the square root (or cube root) of the perfect square (or cube) and multiply it by the remaining radical.

Important Note

Not all radicals can be simplified. If the radicand has no perfect square or cube factors other than 1, the radical is already in its simplest form.

Cube Root Calculator

Our cube root calculator simplifies expressions with cube roots. It's perfect for students, teachers, and anyone working with algebraic expressions.

The calculator uses the following formula to simplify cube roots:

Cube Root Simplification Formula

∛(a × b) = ∛a × ∛b

Where a is a perfect cube and b is the remaining factor.

For example, to simplify ∛108:

Factor 108 into 27 × 4 (since 27 is a perfect cube).
Take the cube root of 27, which is 3.
Multiply by the remaining factor: 3 × ∛4.

The result is 3∛4, which is the simplified form of ∛108.

Examples

Here are some examples of simplified radical expressions:

Square Root Examples

√36 = 6
√80 = 4√5
√128 = 8√2

Cube Root Examples

∛27 = 3
∛125 = 5
∛192 = 4∛3

These examples show how to simplify different types of radical expressions. Use our calculator to verify these results or simplify your own expressions.

Frequently Asked Questions

What is a simplified radical expression?

A simplified radical expression is a radical where the radicand has no perfect square (or cube) factors other than 1. For example, √18 is simplified to 3√2, and ∛54 is simplified to 3∛2.

How do I simplify a cube root?

To simplify a cube root, factor the radicand into a perfect cube and another factor. Then, take the cube root of the perfect cube and multiply it by the remaining radical. For example, ∛108 = 3∛4.

What if the radicand has no perfect square or cube factors?

If the radicand has no perfect square or cube factors other than 1, the radical is already in its simplest form. For example, √7 and ∛11 cannot be simplified further.

Can I simplify radicals with variables?

Yes, you can simplify radicals with variables by factoring out perfect squares (or cubes) from the variable part. For example, √(18x²) = 3x√2, and ∛(27y³) = 3y∛y.

What if I get a negative number inside the radical?

If the radicand is negative, the radical is not a real number. For example, √(-1) and ∛(-8) are not real numbers. You can use imaginary numbers to represent these, but they are beyond the scope of this calculator.