Simplying The Square Root Calculator

Simplifying square roots is a fundamental math skill that helps you express square roots in their simplest radical form. This process involves factoring the radicand (the number under the square root) into perfect squares and simplifying the expression. Our calculator makes this process quick and easy, while our guide explains the steps in detail.

What is simplifying square roots?

Simplifying square roots means expressing a square root in its simplest radical form. This involves factoring the radicand into perfect squares and taking those squares out of the square root. The simplified form typically has a smaller radicand, making the expression easier to work with.

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.

Simplifying square roots is different from rationalizing denominators, which involves eliminating radicals from the denominator of a fraction.

How to simplify square roots

To simplify a square root, follow these steps:

Factor the radicand into perfect squares and other factors.
Take the square root of the perfect squares and write them as coefficients outside the square root.
Leave any remaining factors inside the square root.

√(a × b) = √a × √b

If a is a perfect square, √a = √(n²) = n

For example, to simplify √72:

Factor 72 into 36 × 2 (since 36 is a perfect square).
Take the square root of 36, which is 6.
Write the simplified form as 6√2.

Examples of simplifying square roots

Here are some examples of simplifying square roots:

Original Square Root	Simplified Form	Explanation
√36	6	36 is a perfect square (6×6).
√72	6√2	72 = 36 × 2, and 36 is a perfect square.
√108	6√3	108 = 36 × 3, and 36 is a perfect square.
√192	8√3	192 = 64 × 3, and 64 is a perfect square.

Common mistakes to avoid

When simplifying square roots, it's easy to make a few common mistakes:

Not factoring the radicand completely. Always factor until you can't find any more perfect squares.
Taking the square root of the entire radicand instead of the perfect square factors. For example, √(36 × 2) = 6√2, not √72 = 8.485.
Forgetting to simplify the square root of a perfect square. For example, √16 should be simplified to 4, not left as √16.

Always double-check your work to ensure you've factored the radicand completely and taken the square root of all perfect square factors.

FAQ

What is the difference between simplifying square roots and rationalizing denominators?: Simplifying square roots involves expressing the square root in its simplest radical form by factoring the radicand into perfect squares. Rationalizing denominators involves eliminating radicals from the denominator of a fraction by multiplying the numerator and denominator by the conjugate of the denominator.
Can I simplify a square root that has a variable inside it?: Yes, you can simplify square roots with variables by factoring out perfect square factors. For example, √(18x²) can be simplified to 3x√2.
What if the radicand doesn't have any perfect square factors?: If the radicand doesn't have any perfect square factors other than 1, then the square root is already in its simplest form. For example, √7 cannot be simplified further.
How do I simplify square roots with negative numbers?: The square root of a negative number is not a real number, but it can be expressed as an imaginary number using the imaginary unit i (where i² = -1). For example, √(-4) = 2i.