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Simplify Irrational Square Roots Calculator

Reviewed by Calculator Editorial Team

Simplifying irrational square roots is a fundamental math skill that helps solve equations, work with measurements, and understand algebraic expressions. This guide explains the rules for simplifying √a/b, √(a+b), and other complex roots, with practical examples and a built-in calculator.

How to Simplify Irrational Square Roots

The process of simplifying square roots involves reducing the radical expression to its simplest form where no perfect square factors remain under the radical. Here's the step-by-step approach:

  1. Factor the radicand: Break down the number under the square root into its prime factors.
  2. Identify perfect squares: Look for factors that are perfect squares (like 4, 9, 16, etc.).
  3. Separate the square roots: Move the perfect square factors outside the square root.
  4. Simplify the remaining root: If possible, simplify what remains under the square root.

General simplification formula:

√(a × b) = √a × √b

√(a/b) = √a / √b

√(a + b) cannot be simplified further unless a and b have common factors

For example, √36 simplifies to 6 because 36 is a perfect square. Similarly, √72 simplifies to 6√2 because 72 = 36 × 2 and 36 is a perfect square.

Common Simplification Formulas

Here are the most frequently used formulas for simplifying square roots:

Original Expression Simplified Form Example
√(a × b) √a × √b √(8 × 2) = √8 × √2 = 2√2 × √2 = 2 × 2 = 4
√(a/b) √a / √b √(18/2) = √18 / √2 = 3√2 / √2 = 3
√(a + b) Cannot simplify unless a and b have common factors √(9 + 16) = √25 = 5
√(a - b) Cannot simplify unless a and b have common factors √(25 - 9) = √16 = 4

Important Note: The product and quotient rules only apply when the square roots are multiplied or divided. Addition and subtraction under the square root cannot be simplified further unless the entire radicand is a perfect square.

Worked Examples

Example 1: Simplifying √72

  1. Factor 72: 72 = 36 × 2
  2. 36 is a perfect square (6²)
  3. √72 = √(36 × 2) = √36 × √2 = 6√2

Example 2: Simplifying √(18/2)

  1. √(18/2) = √18 / √2
  2. Factor 18: 18 = 9 × 2
  3. 9 is a perfect square (3²)
  4. √18 = √(9 × 2) = √9 × √2 = 3√2
  5. So √(18/2) = 3√2 / √2 = 3

Example 3: Simplifying √(9 + 16)

  1. 9 + 16 = 25
  2. 25 is a perfect square (5²)
  3. √(9 + 16) = √25 = 5

FAQ

Can I simplify √(a + b) if a and b aren't perfect squares?
No, unless a + b itself is a perfect square. For example, √(9 + 16) simplifies to 5, but √(8 + 2) cannot be simplified further.
What if the radicand has a negative number?
Square roots of negative numbers are not real numbers. For example, √(-1) is an imaginary number (i).
How do I simplify nested square roots like √(√a)?
Nested square roots can sometimes be simplified by expressing them as exponents. For example, √(√a) = a^(1/4).
Can I simplify √(a - b) if a and b have common factors?
Only if a - b is a perfect square. For example, √(25 - 9) simplifies to 4, but √(16 - 8) cannot be simplified further.