Write The Following in Simplified Radical Form Calculator

This calculator helps you convert numbers into simplified radical form. Radical expressions are written with a radical sign (√) and a radicand (the number under the radical). Simplified radical form means the radicand has no perfect square factors other than 1.

What is Simplified Radical Form?

Simplified radical form is a way of writing square roots that makes them as simple as possible. In simplified radical form:

The radicand (the number under the radical) has no perfect square factors other than 1.
The radical is written with the smallest possible radicand.
Any perfect square factors are moved outside the radical as a separate factor.

For example, √36 simplifies to 6 because 36 is a perfect square (6×6). The simplified radical form of √72 would be 6√2 because 72 can be factored into 36×2, and 36 is a perfect square.

How to Simplify Radicals

To simplify a radical expression, follow these steps:

Factor the radicand: Break down the number under the radical into its prime factors.
Identify perfect squares: Look for factors that are perfect squares (like 4, 9, 16, 25, etc.).
Separate the perfect squares: Move any perfect square factors outside the radical.
Simplify the remaining radicand: If there are any perfect square factors left in the radicand, repeat the process.

Formula

√(a × b) = √a × √b

If a is a perfect square, √(a × b) = √a × √b = √a × √b = √a × √b

Note

Remember that only perfect square factors can be moved outside the radical. For example, √18 cannot be simplified further because 18 has no perfect square factors other than 1.

Examples

Let's look at a few examples to see how simplification works:

Original Expression	Simplified Form	Explanation
√36	6	36 is a perfect square (6×6), so the radical simplifies to 6.
√72	6√2	72 can be factored into 36×2. 36 is a perfect square, so we move it outside the radical: √(36×2) = √36 × √2 = 6√2.
√128	8√2	128 can be factored into 64×2. 64 is a perfect square, so we move it outside the radical: √(64×2) = √64 × √2 = 8√2.
√18	3√2	18 can be factored into 9×2. 9 is a perfect square, so we move it outside the radical: √(9×2) = √9 × √2 = 3√2.

FAQ

What is the difference between a radical and a simplified radical?: A radical is any expression with a square root symbol. A simplified radical is one where the radicand has no perfect square factors other than 1, and any perfect square factors have been moved outside the radical.
Can I simplify a radical that has a variable inside it?: Yes, you can simplify radicals with variables by factoring out perfect square factors. For example, √(18x²) can be simplified to 3x√2.
What if the radicand has no perfect square factors?: If the radicand has no perfect square factors other than 1, then the radical is already in its simplest form. For example, √7 cannot be simplified further.
How do I simplify a radical with a negative number?: Radicals with negative numbers can be simplified by factoring out the negative sign. For example, √(-27) can be written as -√27, which can then be simplified to -3√3.