Simplify Radicals Calculator
Result:
Simplify Radical Expressions to their simplest form with our comprehensive calculator. Learn the systematic approach to radical simplification and understand the underlying mathematical principles.
What is Simplifying Radicals?
Simplifying radicals means expressing a radical in its simplest form by factoring out perfect powers and reducing the expression to have the smallest possible radicand.
Steps to Simplify Radicals
General Method
- Find prime factorization of the radicand
- Group factors according to the index
- Extract perfect powers from under the radical
- Multiply extracted factors outside the radical
- Leave remaining factors under the radical
Detailed Example: Simplify √72
Step-by-Step Solution
Step 1: Prime factorization of 72
72 = 8 × 9 = 2³ × 3² = 2² × 2 × 3²
Step 2: Identify perfect squares
72 = 4 × 18 = 2² × 18 (4 is a perfect square)
Or: 72 = 36 × 2 = 6² × 2 (36 is a perfect square)
Step 3: Extract the perfect square
√72 = √(36 × 2) = √36 × √2 = 6√2
Result: √72 = 6√2
Check: (6√2)² = 36 × 2 = 72 ✓
Rules for Different Root Types
Square Roots (index 2)
Look for: Perfect squares (1, 4, 9, 16, 25...)
Factor pairs: Any factor raised to power 2
Examples:
- √50 = √(25×2) = 5√2
- √18 = √(9×2) = 3√2
- √200 = √(100×2) = 10√2
Cube Roots (index 3)
Look for: Perfect cubes (1, 8, 27, 64, 125...)
Factor groups: Any factor raised to power 3
Examples:
- ∛54 = ∛(27×2) = 3∛2
- ∛16 = ∛(8×2) = 2∛2
- ∛250 = ∛(125×2) = 5∛2
Prime Factorization Method
Example: Simplify √300
Step 1: Prime factorization
300 = 2² × 3 × 5² = 4 × 3 × 25
Step 2: Group perfect squares
300 = (2² × 5²) × 3 = (2 × 5)² × 3 = 10² × 3
Step 3: Simplify
√300 = √(10² × 3) = 10√3
Verification: (10√3)² = 100 × 3 = 300 ✓
Perfect Powers Reference
Perfect Squares
1² = 1, 2² = 4, 3² = 9
4² = 16, 5² = 25, 6² = 36
7² = 49, 8² = 64, 9² = 81
10² = 100, 11² = 121, 12² = 144
13² = 169, 14² = 196, 15² = 225
Perfect Cubes
1³ = 1, 2³ = 8, 3³ = 27
4³ = 64, 5³ = 125, 6³ = 216
7³ = 343, 8³ = 512, 9³ = 729
10³ = 1000
Perfect Fourth Powers
1⁴ = 1, 2⁴ = 16, 3⁴ = 81
4⁴ = 256, 5⁴ = 625
6⁴ = 1296
Common Simplification Patterns
Factor of 2
√8 = √(4×2) = 2√2
√18 = √(9×2) = 3√2
√32 = √(16×2) = 4√2
Common pattern with 2Factor of 3
√12 = √(4×3) = 2√3
√27 = √(9×3) = 3√3
√48 = √(16×3) = 4√3
Common pattern with 3Factor of 5
√20 = √(4×5) = 2√5
√45 = √(9×5) = 3√5
√80 = √(16×5) = 4√5
Common pattern with 5Simplifying Higher-Index Radicals
Example: Simplify ∛432
Step 1: Prime factorization
432 = 16 × 27 = 2⁴ × 3³
Step 2: Group by cube factors
432 = 2³ × 2¹ × 3³ = (2 × 3)³ × 2 = 6³ × 2
Step 3: Extract cube root
∛432 = ∛(6³ × 2) = 6∛2
When Radicals Cannot Be Simplified
A radical is in simplest form when:
- No perfect power factors: Radicand has no factors that are perfect powers of the index
- No fractions: No fractions under the radical sign
- No radicals in denominator: Denominator is rationalized
Applications of Simplified Radicals
- Geometry: Simplifying lengths and distances
- Trigonometry: Exact values of trigonometric functions
- Algebra: Solving radical equations
- Physics: Calculations involving square roots
- Engineering: RMS values and signal processing
Practice Problems
√98
Answer: 7√2
98 = 49 × 2∛135
Answer: 3∛5
135 = 27 × 5√180
Answer: 6√5
180 = 36 × 5⁴√162
Answer: 3⁴√2
162 = 81 × 2Find Calculator
Popular Calculators
Other Calculators
