Square Root Calculator Simplest Form

What is a Square Root?

The square root of a number x, denoted as √x, is a value that, when multiplied by itself, gives the original number x. For example, √9 = 3 because 3 × 3 = 9. Square roots can be either exact or irrational, depending on the number.

In mathematics, the square root function is defined for non-negative real numbers and is written as √x. For negative numbers, the square root is not a real number but can be expressed using imaginary numbers (i, where i² = -1).

What is Simplest Form?

A square root is in simplest form when there are no perfect square factors left under the radical. A perfect square is an integer that is the square of another integer (e.g., 1, 4, 9, 16, 25, etc.).

For example, √36 is already in simplest form because 36 is a perfect square (6 × 6). However, √72 is not in simplest form because 72 can be factored into 36 × 2, and 36 is a perfect square.

How to Simplify Square Roots

To simplify a square root, follow these steps:

1. Factor the number under the square root into its prime factors.
2. Identify any perfect square factors.
3. Remove the perfect square factors from the radical and multiply them outside the radical.
4. Combine the results to form the simplified radical expression.

For example, to simplify √72:

1. Factor 72: 72 = 36 × 2
2. 36 is a perfect square (6 × 6).
3. Remove 36 from the radical: √72 = √(36 × 2) = √36 × √2 = 6√2

Worked Examples

Example 1: Simplifying √48

1. Factor 48: 48 = 16 × 3
2. 16 is a perfect square (4 × 4).
3. Simplify: √48 = √(16 × 3) = √16 × √3 = 4√3

Example 2: Simplifying √128

1. Factor 128: 128 = 64 × 2
2. 64 is a perfect square (8 × 8).
3. Simplify: √128 = √(64 × 2) = √64 × √2 = 8√2

Example 3: Simplifying √50

1. Factor 50: 50 = 25 × 2
2. 25 is a perfect square (5 × 5).
3. Simplify: √50 = √(25 × 2) = √25 × √2 = 5√2