Square Root Calculator Radical Form Online

This square root calculator provides accurate results in radical form. Simply enter a number and get the square root expressed as √n. The calculator handles both perfect squares and irrational roots, showing exact radical forms when possible.

What is Square Root?

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square roots of 9 are 3 and -3 because 3×3 = 9 and (-3)×(-3) = 9.

Square roots are fundamental in mathematics, geometry, and many scientific fields. They appear in calculations involving areas, distances, and other measurements.

Square Root Formula

For a non-negative number a, the square root is written as:

√a = b where b × b = a

Principal Square Root

The principal (or non-negative) square root is the non-negative value that satisfies the equation. For example, √9 = 3, not -3.

Irrational Square Roots

When a number doesn't have a perfect square, its square root is irrational. For example, √2 cannot be expressed as a simple fraction and continues infinitely without repeating.

Radical Form Explained

Radical form is a way to express square roots using the radical symbol (√). This form is particularly useful when dealing with numbers that aren't perfect squares.

Example: √18 in Radical Form

18 can be factored into 9 × 2, and since 9 is a perfect square:

√18 = √(9 × 2) = √9 × √2 = 3√2

Simplifying Radicals

Radicals can often be simplified by factoring out perfect squares. This process makes the expression cleaner and easier to understand.

Note: The calculator automatically simplifies results to their simplest radical form when possible.

How to Calculate Square Roots

Calculating square roots manually can be done using several methods, including:

Prime Factorization: Break down the number into its prime factors and pair them.
Long Division Method: A more complex method involving repeated subtraction and division.
Using a Calculator: The quickest and most accurate method for most practical purposes.

Using the Calculator

Our online square root calculator provides instant results. Simply enter your number and click "Calculate" to get the square root in radical form.

Square Root Examples
Number	Square Root (Radical Form)
16	4
25	5
30	√30
50	5√2
100	10

Common Applications

Square roots have numerous practical applications across various fields:

Geometry: Calculating lengths of sides, areas, and diagonals.
Physics: Determining distances, velocities, and other measurements.
Finance: Calculating standard deviations and other statistical measures.
Engineering: Solving equations and designing structures.

Example in Geometry

If you have a right triangle with one leg measuring 5 units and the hypotenuse measuring 13 units, you can find the other leg using the Pythagorean theorem:

a² + b² = c²

5² + b² = 13²

25 + b² = 169

b² = 144

b = √144 = 12 units

Frequently Asked Questions

What is the difference between square root and square?

The square of a number is that number multiplied by itself (e.g., 5² = 25). The square root is the value that, when multiplied by itself, gives the original number (e.g., √25 = 5).

Can square roots be negative?

Yes, square roots can be negative. For example, both 3 and -3 are square roots of 9 because 3×3 = 9 and (-3)×(-3) = 9. However, the principal square root is always non-negative.

How do I simplify √50?

√50 can be simplified by factoring out the perfect square 25: √50 = √(25 × 2) = √25 × √2 = 5√2.

What is the square root of a negative number?

The square root of a negative number is not a real number. In mathematics, it's represented using the imaginary unit i, where √(-1) = i. For example, √(-4) = 2i.