Simplify Expression Square Root Calculator

This calculator helps you simplify square root expressions by applying fundamental algebraic rules. Whether you're studying algebra, preparing for exams, or working on math problems, understanding how to simplify √(a²b) and similar expressions is essential.

How to Use This Calculator

To simplify a square root expression:

Enter the expression you want to simplify in the input field. For example, you might enter √(18a²b).
Click the "Simplify" button to process the expression.
The calculator will display the simplified form of the expression, such as 3ab√(2).
Review the step-by-step simplification process shown below the result.

The calculator follows these key simplification rules:

√(a²) = a
√(ab) = √a × √b
√(a/b) = √a / √b
√(a + b) cannot be simplified further unless a and b are perfect squares

Square Root Simplification Rules

When simplifying square roots, follow these fundamental rules:

Basic Simplification Rules

√(a²) = a (if a is non-negative)
√(ab) = √a × √b
√(a/b) = √a / √b
√(a + b) cannot be simplified further unless a and b are perfect squares

For example, √(36x²) simplifies to 6x because 36 is a perfect square (6²). Similarly, √(8x²y) simplifies to 2xy√(2) because 8 is not a perfect square but can be factored into 4 × 2, where 4 is a perfect square.

Important Notes

All variables under a square root must be non-negative for the expression to be real.
Coefficients inside the square root should be factored into perfect squares and other factors.
Variables with even exponents can be moved outside the square root.

Worked Examples

Example 1: Simple Perfect Square

Simplify √(25x²).

Factor 25 into 5²: √(25x²) = √(5²x²)
Apply the √(a²) rule: √(5²x²) = 5x

The simplified form is 5x.

Example 2: Mixed Coefficient and Variable

Simplify √(18a²b).

Factor 18 into 9 × 2: √(18a²b) = √(9 × 2 × a² × b)
Apply the √(ab) rule: √(9a²b) = √9 × √a² × √b = 3a√(2b)

The simplified form is 3a√(2b).

Example 3: Complex Expression

Simplify √(50x²y³).

Factor 50 into 25 × 2: √(50x²y³) = √(25 × 2 × x² × y³)
Apply the √(ab) rule: √(25x²y³) = √25 × √x² × √y³ = 5xy√(2y)

The simplified form is 5xy√(2y).

Frequently Asked Questions

Can I simplify √(a + b)?

No, unless a and b are perfect squares. For example, √(16 + 9) can be simplified to √25 = 5, but √(16 + 10) cannot be simplified further.

What if the expression has a negative coefficient?

If the coefficient is negative, the square root is not a real number. For example, √(-9) is not a real number.

How do I simplify √(a/b)?

Use the rule √(a/b) = √a / √b. For example, √(8/2) = √8 / √2 = 2√2 / √2 = 2.