Simplify Square Root with Variables Calculator

This calculator helps you simplify square roots with variables, such as √(a²b) or √(x² + 2xy + y²). It follows standard algebraic rules to simplify expressions under the square root symbol.

How to Use This Calculator

Enter the expression you want to simplify in the input field. The calculator will apply algebraic rules to simplify the square root. For example, entering "√(a²b)" will return "a√b".

Note: The calculator assumes all variables are positive and real numbers. Complex numbers are not supported.

Square Root Simplification Rules

When simplifying square roots with variables, follow these key rules:

√(a²b) = a√b
√(a² + 2ab + b²) = a + b
√(a² - 2ab + b²) = |a - b|
√(x² + y²) cannot be simplified further unless there's a relationship between x and y

Formula: √(a²b) = a√b

This rule works because the square root of a squared term (a²) is simply a, and the remaining term (b) is placed under the square root.

Worked Examples

Example 1: √(9x²)

Using the formula √(a²b) = a√b:

Identify a² as 9x² → a = 3x
Remaining term is 1 (since 9x² is already squared)
Simplified form: 3x√1 = 3x

Example 2: √(x² + 2xy + y²)

This is a perfect square trinomial:

Recognize the pattern a² + 2ab + b²
Here, a = x and b = y
Simplified form: x + y

Common Mistakes to Avoid

Assuming √(a + b) = √a + √b - This is incorrect unless a and b are perfect squares
Forgetting to simplify the coefficient - Always simplify the coefficient outside the square root
Ignoring negative signs - Remember that √(a²) = |a|, not just a

Frequently Asked Questions

Can I simplify √(x² + y²)?

No, unless there's a specific relationship between x and y. For example, if x = 3 and y = 4, then √(x² + y²) = 5, but this is a specific case.

What if the expression has a negative sign?

For expressions like √(a² - b²), you can write it as √(a²(1 - (b/a)²)) = a√(1 - (b/a)²), but this doesn't simplify further unless b/a is a known ratio.

How do I simplify √(a² + b² + c²)?

This expression cannot be simplified further unless there's a specific relationship between a, b, and c. For example, if a = 3, b = 4, c = 5, then √(a² + b² + c²) = √(9 + 16 + 25) = √50 = 5√2.