Simplify Radicals Calculator
Simplify radical expressions by extracting perfect square (or nth-power) factors. Converts √72 → 6√2, ∛54 → 3∛2, and handles variables.
What Is the Simplify Radicals Calculator?
Simplifying a radical means writing it in its simplest form by pulling out all perfect-power factors from under the radical sign. For a square root, extract the largest perfect-square factor. For a cube root, extract the largest perfect-cube factor. The result is written as a coefficient times a simplified radical: √72 = √(36×2) = 6√2.
Formula
How to Use
Enter any positive integer radicand and choose the root index (2 for square root, 3 for cube root, etc.). The calculator shows the simplified form, the full prime factorisation, and step-by-step working. Enter a negative radicand with an odd root index for real results (e.g., ∛−8 = −2).
Example Calculation
Simplify √180: Factor 180 = 4 × 9 × 5 = 2² × 3² × 5. Extract perfect squares: √(4 × 9 × 5) = 2 × 3 × √5 = 6√5. The simplified form is 6√5.
Understanding Simplify Radicals
Radical simplification is a fundamental skill in algebra and appears extensively in geometry (Pythagorean theorem results like √50), trigonometry (exact values of trig ratios like sin 45° = √2/2), and calculus (rationalisation of denominators and conjugate techniques).
The algorithm works by prime-factorising the radicand, then grouping prime factors into sets of the index size. Each complete group contributes one factor to the coefficient outside the radical; remaining unpaired factors stay inside.
For example, ∛(54) = ∛(2 × 27) = ∛(2 × 3³) = 3∛2. The 3³ group extracts as a single factor 3, leaving 2 inside.
A special case: if the result has no radical remaining (the radicand was a perfect nth power), the expression evaluates to a rational number. Recognising perfect squares up to 400 (20²) and perfect cubes up to 1000 (10³) is useful mental arithmetic for quick simplification.
Frequently Asked Questions
What does it mean to simplify a radical?
A radical is in simplest form when the radicand (number under the √ sign) has no perfect-square (or nth-power) factors other than 1. For example, √12 is not simplified because 12 = 4×3 and 4 is a perfect square. The simplified form is 2√3.
How do you simplify √72 step by step?
1. Find the prime factorisation: 72 = 2³ × 3². 2. Group into pairs of primes (for square root): 2³ = 2² × 2, so 72 = (2²)(3²)(2). 3. Extract each pair: √(2² × 3² × 2) = 2 × 3 × √2 = 6√2.
What if the radicand is a perfect square?
If the radicand is a perfect square (e.g. √144), the result is a whole number (12) with no radical remaining — the radical has been fully eliminated.
Can you simplify radicals with variables?
Yes, the same principle applies: √(x⁴y³) = x²y√y. This calculator handles integer radicands; for variable expressions, apply the same grouping logic manually.
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