Simplyfing Square Roots Calculator

Square roots can be simplified to make them easier to work with in mathematical problems. This calculator helps you simplify square roots by factoring out perfect squares from the radicand (the number under the square root symbol).

What is simplifying square roots?

Simplifying square roots means expressing a square root in its simplest radical form. This involves factoring out the largest perfect square from the radicand. A perfect square is an integer that is the square of another integer (e.g., 1, 4, 9, 16, 25, etc.).

For example, √36 can be simplified to 6 because 36 is a perfect square (6 × 6). Similarly, √72 can be simplified to 6√2 because 72 = 36 × 2, and 36 is a perfect square.

Simplified square root formula:

√(a × b) = √a × √b, where a is the largest perfect square factor of the radicand.

How to simplify square roots

To simplify a square root, follow these steps:

Factor the radicand into a product of perfect squares and other factors.
Identify the largest perfect square factor.
Take the square root of the perfect square factor.
Multiply this by the square root of the remaining factor.

If the radicand is a perfect square, the simplified form is simply the square root of that number.

Example: Simplify √72

1. Factor 72: 72 = 36 × 2

2. 36 is a perfect square (6 × 6)

3. √36 = 6

4. Final simplified form: 6√2

Examples

Original Square Root	Simplified Form	Explanation
√18	3√2	18 = 9 × 2, √9 = 3
√50	5√2	50 = 25 × 2, √25 = 5
√80	4√5	80 = 16 × 5, √16 = 4
√128	8√2	128 = 64 × 2, √64 = 8

Common mistakes

When simplifying square roots, common errors include:

Not factoring the radicand correctly
Choosing a perfect square factor that is not the largest possible
Forgetting to multiply the square root of the perfect square by the remaining factor
Assuming all radicands can be simplified (some cannot be simplified further)

Tip: Always check if the radicand has any perfect square factors other than 1 before considering the square root simplified.

FAQ

Can all square roots be simplified?

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

What if the radicand is a negative number?

Square roots of negative numbers are not real numbers. They are considered imaginary numbers and are typically expressed with the imaginary unit i (√-1 = i).

How do I simplify square roots with variables?

The process is similar to simplifying numerical square roots. Factor the radicand and look for perfect square factors, including variable terms. For example, √(18x²) = 3x√2.