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Simplifying Adding and Subtracting Square Roots Calculator

Reviewed by Calculator Editorial Team

This guide explains how to simplify expressions involving square roots, including addition and subtraction. We'll cover the rules for combining square roots, provide examples, and show you how to use our calculator to simplify complex expressions.

How to Use This Calculator

Our calculator helps simplify expressions with square roots. Enter your expression in the format √a ± √b, where a and b are positive numbers. The calculator will simplify the expression to its simplest radical form.

For example, if you enter √8 + √18, the calculator will simplify it to 5√2.

Note: The calculator assumes you're working with positive square roots. For negative square roots, you'll need to use the imaginary unit i (√-1 = i).

Simplifying Square Roots

Before adding or subtracting square roots, it's important to simplify them as much as possible. A square root is in its simplest form when:

  • The radicand (the number under the square root) has no perfect square factors other than 1
  • The radical is multiplied by an integer coefficient

To simplify a square root:

  1. Factor the radicand into perfect squares and other factors
  2. Take the square root of the perfect squares
  3. Leave the remaining factors under the radical

√(a × b) = √a × √b

√(a² × b) = a × √b

Example: Simplify √72

  1. Factor 72: 72 = 36 × 2
  2. √72 = √(36 × 2) = √36 × √2 = 6√2

Adding Square Roots

You can add two square roots if they have the same radicand. The rule is:

√a + √a = 2√a

√a + √b cannot be simplified if a ≠ b

Example: Simplify √8 + √18

  1. Simplify each square root:
    • √8 = √(4 × 2) = 2√2
    • √18 = √(9 × 2) = 3√2
  2. Combine like terms: 2√2 + 3√2 = 5√2

Subtracting Square Roots

The same rule applies to subtraction:

√a - √a = 0

√a - √b cannot be simplified if a ≠ b

Example: Simplify √50 - √8

  1. Simplify each square root:
    • √50 = √(25 × 2) = 5√2
    • √8 = √(4 × 2) = 2√2
  2. Subtract: 5√2 - 2√2 = 3√2

Common Mistakes to Avoid

When working with square roots, these common errors often occur:

  • Adding or subtracting square roots with different radicands: √a + √b ≠ √(a + b)
  • Assuming √(a + b) = √a + √b
  • Forgetting to simplify square roots before combining them
  • Incorrectly factoring radicands

Remember: You can only combine square roots when they have identical radicands. Always simplify before adding or subtracting.

FAQ

Can I add square roots with different numbers under them?
No, you can only add square roots when they have the same radicand. For example, √2 + √3 cannot be simplified further.
What if I have a negative number under a square root?
Square roots of negative numbers are not real numbers. In mathematics, √(-1) is defined as the imaginary unit i. For example, √(-4) = 2i.
How do I simplify expressions with variables under square roots?
The same rules apply. Factor the radicand and take the square root of any perfect square factors. For example, √(18x²) = 3x√2.
Can I use this calculator for complex expressions?
Our calculator is designed for basic expressions of the form √a ± √b. For more complex expressions, you may need to simplify them manually first.