Simplifying Radicals Calculator
Simplify square roots and nth roots
Understanding Radical Simplification
Simplifying radicals means expressing a radical expression in its simplest form by extracting perfect powers from under the radical sign. This makes the expression cleaner and often easier to work with.
The Process
- Find the prime factorization of the radicand
- Look for groups of factors that match the index
- Extract one factor for each complete group
- Multiply extracted factors to get the coefficient
- Multiply remaining factors to get the new radicand
Perfect Squares to Memorize
Common Examples
- √72: 72 = 36 × 2 = 6²× 2, so √72 = 6√2
- √50: 50 = 25 × 2 = 5² × 2, so √50 = 5√2
- √48: 48 = 16 × 3 = 4² × 3, so √48 = 4√3
- ∛54: 54 = 27 × 2 = 3³ × 2, so ∛54 = 3∛2
Why Simplify Radicals?
- Makes addition and subtraction of like radicals possible
- Easier to identify and combine like terms
- Standard mathematical convention
- Simpler to compare radical values
- Required format for most standardized tests
Applications
- Geometry: Diagonal lengths and triangle calculations
- Physics: Velocity and acceleration formulas
- Engineering: Stress analysis and structural calculations
- Statistics: Standard deviation calculations
- Computer Graphics: Distance and vector calculations
Frequently Asked Questions
What does it mean to simplify a radical?
Simplifying a radical means removing all perfect powers from under the radical sign. For square roots, you look for perfect squares; for cube roots, perfect cubes, etc. The goal is to make the number under the radical as small as possible while moving factors outside.
How do I know if a radical is fully simplified?
A radical is fully simplified when: (1) No perfect powers remain under the radical, (2) No fractions are under the radical, (3) No radicals appear in the denominator of a fraction. For square roots, this means no factor under the radical should be divisible by a perfect square.
What's the fastest way to simplify radicals?
Look for the largest perfect power that divides the radicand. For square roots, check if the number is divisible by 4, 9, 16, 25, 36, 49, etc. If you find one, divide it out and take its root. For example, if you see 72, notice it's divisible by 36, so √72 = √(36×2) = 6√2.
Can all radicals be simplified?
No, not all radicals can be simplified. Prime numbers under a radical (like √2, √3, √5) are already in simplest form because they have no perfect power factors other than 1. These are called "simplified radicals" or "radicals in simplest form."
What if I can't find the prime factorization easily?
Start by testing divisibility by small primes: 2, 3, 5, 7, 11, etc. Or look for recognizable perfect squares (4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144). Even partial simplification helps - you don't always need the complete prime factorization right away.
How do I simplify radicals with variables?
Apply the same principle to variables. For √(x⁸), since 8 is divisible by 2 (the index), you can take out x⁴, giving you x⁴. For √(x⁷), you can take out x³ (6 factors), leaving x under the radical: x³√x. Each pair of identical factors comes out as one factor.