Square Root Calculator Exact Form

Finding exact square roots is essential in mathematics, engineering, and science. This calculator helps you determine the exact form of square roots, which is often more useful than decimal approximations.

What is a 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 can be either exact or approximate. Exact square roots are expressed in radical form (√), while approximate square roots are decimal numbers.

Exact Form of Square Roots

The exact form of a square root is expressed as a simplified radical. This means the number inside the square root (the radicand) is broken down into its prime factors, and any perfect squares are taken out of the radical.

Exact Form Formula:

√a = √(b² × c) = b√c, where b² is the largest perfect square factor of a, and c is the remaining factor.

For example, √50 can be simplified to 5√2 because 50 = 25 × 2 and 25 is a perfect square.

How to Calculate Exact Square Roots

Step-by-Step Process

Identify the largest perfect square that divides the radicand.
Divide the radicand by this perfect square.
Take the square root of the remaining factor.
Multiply the square root of the perfect square by the square root of the remaining factor.

Example Calculation

Let's find the exact form of √72:

The largest perfect square that divides 72 is 36 (6 × 6).
Divide 72 by 36 to get 2.
√36 = 6 and √2 = √2.
Multiply them together: 6√2.

The exact form of √72 is 6√2.

Examples of Exact Square Roots

Number	Exact Form
√8	2√2
√18	3√2
√20	2√5
√50	5√2
√128	8√2

FAQ

What is the difference between exact and approximate square roots?: Exact square roots are expressed in radical form and are precise, while approximate square roots are decimal numbers that are rounded to a certain number of decimal places.
How do I simplify a square root?: To simplify a square root, factor the radicand into perfect squares and non-perfect squares, then take the square root of the perfect squares out of the radical.
Can I find the exact form of any square root?: Yes, you can find the exact form of any square root, but it may not always be simpler than the original radical form.
What is the exact form of √1?: The exact form of √1 is simply 1, since 1 × 1 = 1.
How do I know if a square root is in its simplest form?: A square root is in its simplest form when the radicand has no perfect square factors other than 1.