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Online Simplify Square Root Calculator

Reviewed by Calculator Editorial Team

Simplifying square roots is a fundamental math skill that helps you express radicals in their most reduced form. This calculator makes it easy to simplify square roots of integers and fractions. Whether you're studying algebra, chemistry, or just need a quick reference, this tool will help you understand and work with square roots more efficiently.

How to Use the Calculator

Using our online simplify square root calculator is straightforward. Follow these simple steps:

  1. Enter the number you want to simplify under the square root in the input field.
  2. Click the "Calculate" button to see the simplified form of the square root.
  3. Review the result and the step-by-step explanation of how the simplification was achieved.
  4. If needed, use the "Reset" button to clear the input and start over.

The calculator will display the simplified square root in its simplest radical form, which means the radicand (the number under the square root) has no perfect square factors other than 1.

How Simplifying Square Roots Works

Simplifying a square root involves breaking down the radicand into its prime factors and then separating the perfect squares from the remaining factors. Here's how the process works:

  1. Factor the radicand into its prime factors.
  2. Identify any perfect square factors (numbers that are squares of integers).
  3. Take the square root of the perfect square factors and move them outside the square root.
  4. Combine the remaining factors under the square root to form the simplified radical.

Formula: √a = √(b² × c) = b × √c, where b² is the largest perfect square factor of a.

For example, to simplify √72:

  1. Factor 72: 72 = 36 × 2 = 6² × 2
  2. Take the square root of 36: √36 = 6
  3. Combine: √72 = 6 × √2

Examples of Simplified Square Roots

Here are some examples of simplified square roots to help you understand the process:

Original Square Root Simplified Form Explanation
√16 4 16 is a perfect square (4²)
√50 5√2 50 = 25 × 2 = 5² × 2
√80 4√5 80 = 16 × 5 = 4² × 5
√128 8√2 128 = 64 × 2 = 8² × 2

These examples show how to simplify square roots by factoring out the largest perfect square from the radicand.

Frequently Asked Questions

What is a simplified square root?
A simplified square root is a radical expression where the radicand (the number under the square root) has no perfect square factors other than 1. This means the square root is expressed in its most reduced form.
Can I simplify square roots of fractions?
Yes, you can simplify square roots of fractions by simplifying the numerator and denominator separately. For example, √(8/2) = √4/√2 = 2/√2 = √2 (after rationalizing the denominator).
What if the radicand is a prime number?
If the radicand is a prime number, the square root cannot be simplified further because prime numbers have no perfect square factors other than 1. For example, √7 is already in its simplest form.
How do I simplify nested square roots?
To simplify nested square roots, you can use the property √(√a) = a^(1/4). For example, √(√16) = 16^(1/4) = 2. You can also simplify by expressing the inner square root in its simplest form first.
What if the radicand is negative?
Square roots of negative numbers are not real numbers. In the real number system, the square root of a negative number is undefined. However, in complex numbers, √(-a) = i√a, where i is the imaginary unit.