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Square Root Simplfy Calculator Using Radicals

Reviewed by Calculator Editorial Team

This calculator helps you simplify square roots using radicals. Whether you're working with simple square roots like √18 or more complex expressions like √(72/32), this tool will guide you through the simplification process step by step.

How to Use This Calculator

Using our square root simplifier is straightforward:

  1. Enter the number you want to simplify in the input field.
  2. Click the "Calculate" button to see the simplified radical form.
  3. Review the step-by-step explanation of how the simplification was performed.
  4. Use the "Reset" button to clear the calculator for a new calculation.

The calculator will display the simplified form of the square root and show you the exact steps used to simplify it.

Simplifying Square Roots

Simplifying square roots involves expressing them in the form √(a*b) where a is a perfect square and b has no perfect square factors. Here's how to do it:

Formula

√(a*b) = √a * √b

Where a is the largest perfect square factor of the radicand (the number under the square root).

Steps to Simplify a Square Root

  1. Factor the radicand into perfect squares and other factors.
  2. Separate the square root of the perfect square from the other factors.
  3. Simplify the square root of the perfect square.
  4. Combine the simplified terms.

Important Notes

  • Only factors that are perfect squares can be moved outside the square root.
  • The simplified form should have no perfect square factors remaining under the radical.
  • If the radicand is a perfect square, the square root will simplify to an integer.

Examples

Let's look at a few examples to see how square root simplification works.

Example 1: Simplifying √18

  1. Factor 18: 18 = 9 × 2
  2. 9 is a perfect square (3²), so √9 = 3
  3. √18 = √(9×2) = √9 × √2 = 3√2

Example 2: Simplifying √50

  1. Factor 50: 50 = 25 × 2
  2. 25 is a perfect square (5²), so √25 = 5
  3. √50 = √(25×2) = √25 × √2 = 5√2

Example 3: Simplifying √(72/32)

  1. Simplify the fraction inside the square root: 72/32 = 9/4
  2. Now simplify √(9/4) = √9 / √4 = 3/2

FAQ

What is the difference between simplifying √a and √(a/b)?

Simplifying √a involves factoring a into perfect squares and other factors. Simplifying √(a/b) requires simplifying both the numerator and denominator separately before combining them.

Can I simplify square roots of negative numbers?

Yes, but the result will be an imaginary number. For example, √(-4) = 2i, where i is the imaginary unit.

What if the radicand has no perfect square factors?

The square root is already in its simplest form. For example, √7 cannot be simplified further because 7 has no perfect square factors other than 1.