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Square Root Simplified Radical Form Calculator

Reviewed by Calculator Editorial Team

Simplifying square roots to their radical form is a fundamental math skill that helps in algebra, calculus, and many other areas of mathematics. This calculator makes it easy to simplify square roots quickly and accurately.

What is Simplified Radical Form?

Simplified radical form refers to expressing a square root in its most reduced form, where the radicand (the number under the square root) has no perfect square factors other than 1. For example, √36 simplifies to 6 because 36 is a perfect square (6²).

When simplifying square roots of non-perfect squares, we look for the largest perfect square that divides the radicand. For instance, √72 can be simplified by recognizing that 36 is the largest perfect square that divides 72 (72 ÷ 36 = 2).

Formula

To simplify √a:

  1. Factor a into its prime factors
  2. Pair the prime factors into perfect squares
  3. Take one factor from each pair out of the square root
  4. Multiply the factors outside the square root

How to Simplify Square Roots

Simplifying square roots involves several steps:

  1. Factor the radicand: Break down the number under the square root into its prime factors.
  2. Identify perfect squares: Look for pairs of identical prime factors that form perfect squares.
  3. Separate the square root: Take one factor from each perfect square pair out of the square root.
  4. Multiply the factors: Multiply the factors taken out of the square root.

Important Note

Only perfect square factors can be moved out of the square root. For example, in √50, 25 is a perfect square factor (5²), so √50 simplifies to 5√2.

Using the Calculator

Our square root simplified radical form calculator makes it easy to simplify square roots. Simply enter the number you want to find the square root of, and the calculator will provide the simplified radical form.

The calculator follows these steps:

  1. Checks if the input is a perfect square
  2. If not, factors the radicand
  3. Identifies perfect square factors
  4. Simplifies the expression

Examples

Let's look at a few examples of simplifying square roots:

Example 1: √32

32 can be factored into 16 × 2, where 16 is a perfect square (4²). So, √32 simplifies to 4√2.

Example 2: √80

80 can be factored into 16 × 5, where 16 is a perfect square (4²). So, √80 simplifies to 4√5.

Example 3: √108

108 can be factored into 36 × 3, where 36 is a perfect square (6²). So, √108 simplifies to 6√3.

FAQ

What is the difference between simplified radical form and decimal form?

Simplified radical form expresses the square root as a product of a perfect square and a radical, while decimal form gives an approximate numerical value. For example, √2 is approximately 1.414 in decimal form but remains √2 in simplified radical form.

Can all square roots be simplified?

Yes, all square roots can be simplified to some extent. If the radicand is a perfect square, the square root simplifies to an integer. If not, it simplifies to a product of an integer and a square root of a non-perfect square.

What if the radicand has no perfect square factors?

If the radicand has no perfect square factors other than 1, the square root is already in its simplest form. For example, √7 cannot be simplified further because 7 has no perfect square factors.