Simplify Square Root Calculator Mathway

Simplifying square roots is a fundamental math skill that helps you express square roots in their most reduced form. This calculator follows the Mathway approach to simplify square roots by factoring out perfect squares from the radicand (the number inside the square root).

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 an integer that is the square of another integer (1, 4, 9, 16, 25, 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.

Why Simplify Square Roots?

Simplified square roots are easier to work with in calculations, comparisons, and further mathematical operations. They provide a more compact and precise representation of the original square root.

How to Simplify Square Roots

Follow these steps to simplify any square root:

Factor the radicand into a product of perfect squares and other factors.
Identify the largest perfect square factor.
Take the square root of the perfect square factor.
Multiply this square root by the remaining factors inside the square root.

For example, to simplify √72:

Factor 72: 72 = 36 × 2
36 is a perfect square (6×6)
√36 = 6
6√2 is the simplified form

Simplify Square Root Formula

General Formula

√(a×b) = √a × √b

Where a is the largest perfect square factor of b

The process involves breaking down the radicand into its prime factors and then grouping them into pairs to identify perfect squares.

Simplify Square Root Examples

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

Simplify Square Root FAQ

What is the difference between simplifying and rationalizing a square root?

Simplifying a square root means expressing it in terms of the largest perfect square factor. Rationalizing involves eliminating square roots from the denominator of a fraction. These are different processes with different goals.

Can I simplify a square root that has variables?

Yes, the same principles apply. You look for perfect square factors in the variable expression. For example, √(18x²) can be simplified to 3x√2.

What if the radicand doesn't have any perfect square factors?

If the radicand is a prime number or doesn't have any perfect square factors other than 1, then the square root is already in its simplest form.