Simplify Radicals Calculator
Simplify square roots, cube roots, and nth roots with detailed step-by-step solutions
Common Examples
√72 = 6√2
√48 = 4√3
∛54 = 3∛2
∛128 = 4∛2
How to Simplify Radicals
Simplifying radicals involves factoring out perfect powers from the radicand. The goal is to express the radical in its simplest form by removing all perfect square factors (for square roots), perfect cube factors (for cube roots), or perfect nth power factors (for nth roots).
Steps to Simplify Radicals
- Find the prime factorization of the radicand
- Group the prime factors in sets of the index number
- For each complete group, take one factor outside the radical
- Multiply the factors outside and inside the radical separately
- Write the final simplified form
Example: Simplify √72
- Prime factorization: 72 = 2 × 2 × 2 × 3 × 3
- Group pairs: (2 × 2) × (3 × 3) × 2
- Extract perfect squares: 2 × 3 = 6 outside, 2 remains inside
- Result: √72 = 6√2
Frequently Asked Questions
What does it mean to simplify a radical?
Simplifying a radical means expressing it in its simplest form by factoring out all perfect powers from the radicand. A radical is fully simplified when no perfect square factors (for square roots) or perfect nth power factors remain under the radical.
How do you know if a radical is simplified?
A radical is simplified when: (1) there are no perfect square factors under a square root, (2) there are no fractions under the radical, (3) there are no radicals in the denominator, and (4) the index and exponent have no common factors other than 1.
What is the difference between √ and ∛?
The √ symbol represents a square root (index 2), which asks "what number times itself equals the radicand?" The ∛ symbol represents a cube root (index 3), which asks "what number cubed equals the radicand?"
Can you simplify √50?
Yes! √50 = √(25 × 2) = √25 × √2 = 5√2. We factored out the perfect square 25 and took its square root (5) outside the radical, leaving 2 under the radical.
What if the radicand is a prime number?
If the radicand is a prime number, the radical is already in simplest form because prime numbers have no factors other than 1 and themselves. For example, √7 cannot be simplified further.
How do you simplify cube roots?
For cube roots, find and factor out perfect cubes instead of perfect squares. For example, ∛54 = ∛(27 × 2) = ∛27 × ∛2 = 3∛2, because 27 is a perfect cube (3³ = 27).
Can you simplify radicals with variables?
Yes! For variables, divide the exponent by the index. For example, √(x⁶) = x³ because 6 ÷ 2 = 3. If there's a remainder, it stays under the radical: √(x⁷) = x³√x.
What are perfect squares and perfect cubes?
Perfect squares are numbers that result from squaring an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc. Perfect cubes are numbers from cubing an integer: 1, 8, 27, 64, 125, 216, etc.
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