Square Root Calculator — Simplify Radicals & Perfect Squares

How to Calculate a Square Root

To calculate the square root of any positive number, follow these simple steps:

1. Enter a positive number — Type in the integer or decimal value (e.g., 144).
2. Click “Calculate Square Root” — The tool will calculate the square root value, tell you if the number is a perfect square, and calculate the square of the input number.
3. Read the step-by-step simplification — If the square root is not a whole number, the tool will simplify the radical (e.g., √50 = 5√2).

What is a Square Root?

In mathematics, the square root of a number x is a number y such that y² = x. In other words, a number that, when multiplied by itself, yields the original number.

For example, the square root of 9 is 3, because 3 × 3 = 9. We write this as √9 = 3.

Every positive number has two square roots: one positive and one negative (e.g., √9 is +3 and -3). In mathematics and general practice, the radical symbol √ refers to the principal square root, which is the positive root.

Perfect Squares vs. Irrational Square Roots

Square roots can be classified into two categories based on their results:

• Perfect Squares: Integers that have whole number square roots. Examples include: 1 (√1 = 1), 4 (√4 = 2), 9 (√9 = 3), 16 (√16 = 4), 25 (√25 = 5), 100 (√100 = 10), and 144 (√144 = 12).
• Irrational Numbers: Numbers that do not have whole number square roots. Their roots are decimals that continue infinitely without repeating. For example, √2 ≈ 1.414213… and √3 ≈ 1.732050… These are irrational numbers.

Practical Calculation Examples

Example 1: Perfect Square

Find the square root of 64. Since 8 multiplied by 8 is 64, √64 = 8. It is a perfect square.

Example 2: Simplifying √75 (Radical Form)

To simplify √75, look for perfect square factors of 75. 75 can be divided by 25 (a perfect square: 5² = 25). So, √75 = √25 × √3 = 5√3. The simplified radical form is 5√3, which is approximately 8.660254.

Example 3: Finding the area of a square

If you have a square floor with an area of 100 square feet, the length of one side is the square root of the area: √100 = 10 feet.

Frequently Asked Questions

Can a negative number have a square root?

In the real number system, negative numbers do not have square roots because any number multiplied by itself is positive (e.g., +2 × +2 = 4 and -2 × -2 = 4). However, in the complex number system, negative numbers have imaginary roots (represented by i, where i = √-1). For example, √-4 = 2i.

How do I calculate a square root manually?

For perfect squares, guess-and-test or prime factorization works best. For non-perfect squares, you can use the long division method or Newton’s approximation method (the Babylonian method), which iteratively refines an estimate: New Guess = (Old Guess + (Number / Old Guess)) ÷ 2.

What is the difference between square root and cube root?

A square root asks “what number multiplied by itself twice equals this number?”. A cube root asks “what number multiplied by itself three times equals this number?” (e.g., the cube root of 8 is 2, since 2 × 2 × 2 = 8). Cube roots are written with a small index: ³√.

What is an radical?

The radical is the symbol √ used to denote square roots or higher-order roots. The number inside the radical symbol is called the radicand.