Simplify Expression with Square Roots Calculator

This calculator helps you simplify expressions containing square roots by applying fundamental algebraic rules. Whether you're studying algebra, preparing for exams, or working on math problems, understanding how to simplify square roots is essential.

How to Use This Calculator

Enter your expression in the input field below. The calculator will simplify the expression by:

Removing square roots from the denominator
Combining like terms under the same square root
Simplifying fractions involving square roots
Rationalizing denominators when possible

The calculator will display the simplified form of your expression along with a step-by-step explanation of how it was achieved.

Square Root Simplification Rules

To simplify expressions with square roots, follow these fundamental rules:

√(a·b) = √a · √b √(a/b) = √a / √b √(a²) = a (when a ≥ 0) √(a + b) cannot be simplified further unless a and b are perfect squares

When simplifying expressions with square roots, always:

Remove radicals from denominators
Combine like terms under the same radical
Simplify any fractions involving radicals
Rationalize denominators when possible

Worked Examples

Example 1: Simple Square Root

Original expression: √(27)

Simplified form: 3√3

Explanation: 27 can be factored into 9 × 3, and √9 = 3.

Example 2: Fraction with Square Root

Original expression: √(18/50)

Simplified form: 3√2 / 5√2

Explanation: First simplify the fraction inside the square root to 9/25, then take the square root of each part.

Example 3: Complex Expression

Original expression: √(50) + √(18) - √(8)

Simplified form: 5√2 + 3√2 - 2√2 = 6√2

Explanation: Each term is simplified individually, then combined.

Common Mistakes to Avoid

When simplifying square roots, avoid these common errors:

Assuming √(a + b) = √a + √b - this is incorrect unless a and b are perfect squares
Forgetting to rationalize denominators in expressions like 1/√2
Not simplifying the radicand (the number inside the square root) completely
Ignoring the domain restrictions (square roots of negative numbers are not real)

Remember: The radicand must be non-negative for real square roots. Complex numbers are beyond the scope of this calculator.

Frequently Asked Questions

Can this calculator simplify cube roots?: No, this calculator is specifically designed for simplifying square roots. For cube roots, you would need a different tool.
What if my expression has variables?: The calculator can handle expressions with variables, but it will only simplify the numerical coefficients and perfect square factors.
How do I simplify nested square roots?: Nested square roots like √(√x) can be simplified by combining the exponents: √(√x) = x^(1/4).
Is there a limit to how complex an expression I can simplify?: The calculator can handle reasonably complex expressions, but very large or nested expressions might not simplify completely.
Can I use this calculator for homework or exams?: Yes, you can use this calculator to check your work and understand the simplification process, but always cite the tool you used.