How do you know when a radical is in simplest form?

Determining if a radical is in its simplest form involves a few key steps. A radical is considered simplified when there are no perfect squares (or higher powers, depending on the root) left under the radical sign, and no fractions exist inside the radical.

Steps to Simplify a Radical

• Identify Perfect Squares: Look for numbers that are perfect squares, such as 1, 4, 9, 16, etc., under the radical.
• Factor the Radicand: Break down the number under the radical into its prime factors.
• Extract Perfect Squares: For each pair of identical factors, you can take one out of the radical. For example, √(4) becomes 2.
• Check for Fractions: Ensure there are no fractions inside the radical. If there are, simplify them first.

Example

Consider the radical √(72). To simplify:

• Factor 72: 72 = 36 × 2 = 6² × 2.
• Extract the perfect square: √(72) = √(36 × 2) = √(36) × √(2) = 6√(2).

Now, 6√(2) is in its simplest form because there are no perfect squares left under the radical.

Final Thoughts

Always remember to check for any remaining factors that can be simplified further. If no perfect squares or fractions remain, your radical is in its simplest form!