Free Simplify Radical Expressions Calculator | Radical Simplifier
Simplified Form
2√3
Coefficient
2
Remaining Radicand
3
How It Works
Find Perfect Squares
Look for perfect square factors in the number
Extract Factors
Pull out perfect squares from under the radical
Common Examples
√12
Perfect square factor: 4
= 2√3
√18
Perfect square factor: 9
= 3√2
√50
Perfect square factor: 25
= 5√2
√72
Perfect square factor: 36
= 6√2
Formula & Step-by-Step Process
Radical Simplification Formula
√(a² × b) = a√b
Where a² is the largest perfect square factor of the original number
Step-by-Step Process
- 1. Find all factors of the number under the radical
- 2. Identify the largest perfect square factor
- 3. Extract the square root of the perfect square
- 4. Write as coefficient × √(remaining factor)
Example: √48
Step 1: 48 = 16 × 3
Step 2: 16 is a perfect square (4²)
Step 3: √16 = 4
Step 4: √48 = 4√3
Calculation Table
| Original | Perfect Square Factor | Remaining Factor | Simplified Form |
|---|---|---|---|
| √8 | 4 | 2 | 2√2 |
| √20 | 4 | 5 | 2√5 |
| √32 | 16 | 2 | 4√2 |
| √45 | 9 | 5 | 3√5 |
| √98 | 49 | 2 | 7√2 |
What is Radical Simplification?
What
Radical simplification is the process of extracting perfect square factors from under a radical sign to make expressions easier to work with.
Why
Simplified radicals are easier to add, subtract, multiply, and compare. They also reveal the true structure of mathematical expressions.
Applications
Algebra, geometry, trigonometry, calculus, physics calculations, and engineering problem-solving.
Frequently Asked Questions
What is a radical expression?
A radical expression contains a radical symbol (√) with a number or expression underneath it. The number under the radical is called the radicand.
How do you simplify a radical?
Find the largest perfect square factor of the radicand, extract its square root as a coefficient, and leave the remaining factor under the radical.
What are perfect squares?
Perfect squares are numbers that result from squaring an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc.
When is a radical fully simplified?
A radical is fully simplified when no perfect square factors remain under the radical sign except for 1.
Can you simplify √7 or √11?
No, because 7 and 11 are prime numbers with no perfect square factors other than 1. They are already in simplest form.