Roots in Simplest Form Calculator

This calculator helps you simplify square roots, cube roots, and other radicals to their simplest form. Learn how to simplify roots step by step with our guide and examples.

What is simplest form for roots?

A root in simplest form is a radical expression where the radicand (the number under the radical) has no perfect square factors other than 1. For square roots, this means the radicand should not be divisible by any perfect square other than 1.

For example, √18 can be simplified to 3√2 because 18 = 9 × 2 and 9 is a perfect square. The simplified form has the smallest possible radicand with no perfect square factors.

Key Concepts

Simplest form means no perfect square factors remain under the radical
Square roots can be simplified by factoring the radicand
Cube roots can be simplified by factoring for perfect cubes
The simplified form has the largest possible perfect power factor outside the radical

How to simplify roots

To simplify a square root:

Factor the radicand into perfect squares and other factors
Take the square root of the perfect square factors
Leave the remaining factors under the radical
Combine the results

Example: Simplify √72

1. Factor 72: 72 = 36 × 2 (since 36 is a perfect square)

2. √36 = 6

3. Remaining factor is 2

4. Final simplified form: 6√2

For cube roots, the process is similar but looks for perfect cubes:

Factor the radicand into perfect cubes and other factors
Take the cube root of the perfect cube factors
Leave the remaining factors under the radical
Combine the results

Important Notes

Only perfect squares (for square roots) or perfect cubes (for cube roots) can be moved outside the radical
All factors must be integers for simplification to be possible
Negative radicands can be simplified by factoring out -1
Variables in radicals can also be simplified using the same principles

Examples of simplified roots

Original Root	Simplified Form	Explanation
√50	5√2	50 = 25 × 2, √25 = 5
√108	6√3	108 = 36 × 3, √36 = 6
√128	8√2	128 = 64 × 2, √64 = 8
∛27	3	27 is a perfect cube (3³)
∛108	3∛4	108 = 27 × 4, ∛27 = 3

Frequently Asked Questions

Can all square roots be simplified?

No, only square roots with radicands that have perfect square factors can be simplified. For example, √2 cannot be simplified because 2 has no perfect square factors other than 1.

How do I simplify a negative square root?

First factor out -1 from the radicand, then simplify the positive part. For example, √(-24) = √(-1 × 24) = √(-1) × √24 = i × 2√6.

What's the difference between simplifying square roots and cube roots?

Square roots look for perfect square factors (like 4, 9, 16), while cube roots look for perfect cube factors (like 8, 27, 64). The process is otherwise identical.

Can I simplify roots with variables?

Yes, the same principles apply. For example, √(18x²) = 3x√2, and ∛(27x³) = 3x∛x.