Simplify Square Root of Calculator

Simplifying square roots is a fundamental math skill that helps you express square roots in their most reduced form. This calculator makes it easy to simplify square roots of any integer, and this guide explains exactly how it works.

What is Simplify Square Root?

Simplifying a square root means expressing it in the form √(a×b) where a is the largest perfect square that divides b. A perfect square is a number that is the square of an integer (like 1, 4, 9, 16, etc.).

For example, √36 can be simplified to 6 because 36 is a perfect square (6×6). Similarly, √72 can be simplified to 6√2 because 72 = 36 × 2, and 36 is a perfect square.

Formula: √(a×b) = √a × √b = a × √b (when a is a perfect square)

How to Simplify Square Roots

Follow these steps to simplify any square root:

Factor the number under the square root into a product of perfect squares and other factors.
Separate the square root of the perfect square from the other factors.
Simplify the square root of the perfect square to its integer value.
Combine the simplified perfect square with the remaining factors under one square root.

Example:

Simplify √72

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

Final simplified form: 6√2

Examples of Simplifying Square Roots

Original Square Root	Simplified Form	Explanation
√18	3√2	18 = 9 × 2, √9 = 3
√50	5√2	50 = 25 × 2, √25 = 5
√108	6√3	108 = 36 × 3, √36 = 6
√192	8√3	192 = 64 × 3, √64 = 8

Common Mistakes

When simplifying square roots, it's easy to make these common errors:

Not factoring the number completely - you must find the largest perfect square factor.
Forgetting to simplify the square root of the perfect square to its integer value.
Moving factors outside the square root that aren't perfect squares.
Incorrectly combining terms under the square root.

Tip: Always double-check your factorization and simplification steps to ensure accuracy.

FAQ

Can I simplify square roots of decimals or fractions?: This calculator works best with integers. For decimals or fractions, you may need to convert them to whole numbers first.
What if the number under the square root isn't a perfect square?: The calculator will still simplify it as much as possible by finding the largest perfect square factor.
Is there a difference between simplifying √a and √(a×b)?: Yes. √a is already in its simplest form unless a is a perfect square. √(a×b) can be simplified if a is a perfect square.
Can I simplify square roots of negative numbers?: This calculator only works with positive numbers. For negative numbers, you'll need to use imaginary numbers.