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Simplify Square Root Addition or Subtraction Calculator

Reviewed by Calculator Editorial Team

Adding or subtracting square roots can be simplified using basic algebraic principles. This calculator helps you combine or subtract square roots with like radicands (the numbers under the square root). Learn the rules and practice with examples.

How to Simplify Square Root Addition or Subtraction

Square roots can be added or subtracted when they have the same radicand. Here's how to simplify expressions like √a + √a or √a - √a:

Key Rule

√a + √a = 2√a

√a - √a = 0

Step-by-Step Process

  1. Identify the radicands of both square roots. They must be identical for simplification.
  2. If the radicands are the same, combine the coefficients (numbers in front of the square roots).
  3. Multiply the combined coefficient by the square root of the radicand.
  4. For subtraction, ensure the result isn't negative.

Common Mistakes to Avoid

  • Trying to add or subtract square roots with different radicands (e.g., √2 + √3 cannot be simplified further).
  • Forgetting to combine coefficients properly.
  • Assuming √a² = a (it's actually |a|).

Formula Used

For expressions with the same radicand: n√a ± m√a = (n ± m)√a

Where:

  • n and m are coefficients (numbers in front of the square roots)
  • a is the radicand (the number under the square root)

Worked Examples

Example 1: Adding Square Roots

Simplify 3√5 + 2√5

  1. Identify radicands: Both have √5
  2. Combine coefficients: 3 + 2 = 5
  3. Result: 5√5

Example 2: Subtracting Square Roots

Simplify 7√8 - 4√8

  1. Identify radicands: Both have √8
  2. Combine coefficients: 7 - 4 = 3
  3. Result: 3√8

Example 3: Different Radicands

Simplify √2 + √3

  1. Identify radicands: √2 and √3 are different
  2. Cannot be simplified further
  3. Result: √2 + √3

FAQ

Can I add or subtract square roots with different radicands?

No, square roots can only be added or subtracted when they have the same radicand. For example, √2 + √3 cannot be simplified further.

What happens when I subtract a larger square root from a smaller one?

The result will be negative. For example, 2√5 - 3√5 = -√5. This is mathematically valid but may not be meaningful in all contexts.

Can I simplify expressions like √(a + b) + √(a - b)?

No, these expressions cannot be simplified using basic algebraic rules. They would require more advanced techniques like rationalizing or expanding.