Radical Calculator - Simplify Square, Cube & Nth Roots
Simplify any radical expression — square roots, cube roots, and nth roots — by factoring out perfect powers. Get the simplified radical form instantly.
Enter a radicand (the number inside the root) and the index (degree of the root), then click Simplify Radical to see the simplified form.
Radical Calculator - Simplify Square, Cube & Nth Roots
Simplify any radical expression — square roots, cube roots, and nth roots — by factoring out perfect powers. Get the simplified radical form instantly.
About the Radical Calculator
A radical expression consists of a radical sign (√), a radicand (the number under the sign), and optionally an index (a small number written in the upper-left of the radical sign indicating the degree of the root). The square root is the most familiar radical — it has an implied index of 2. A cube root has index 3, a fourth root has index 4, and so on. The general notation ⁿ√a means the nth root of a, i.e., the number b such that bⁿ = a.
Simplifying a radical means rewriting it so that no perfect nth powers remain under the radical sign. The technique relies on the product rule for radicals: ⁿ√(a·b) = ⁿ√a · ⁿ√b. To simplify ⁿ√a, you factor a into a perfect nth power times a remaining factor. For example, to simplify √50, note that 50 = 25 × 2 = 5² × 2, so √50 = √(5²) · √2 = 5√2.
For higher-order roots the same principle applies but we look for perfect nth power factors. To simplify ∛54 (cube root of 54), factor 54 = 27 × 2 = 3³ × 2, so ∛54 = ∛(3³) · ∛2 = 3∛2. For a fifth root, ⁵√96 = ⁵√(32 × 3) = ⁵√(2⁵ × 3) = 2·⁵√3.
This radical calculator performs this factoring algorithm automatically. It finds the largest perfect nth power factor of the radicand and extracts its nth root as the coefficient, leaving the remaining factor under the radical sign. If the radicand itself is a perfect nth power, the result is a whole number with no radical remaining.
Radical simplification is essential in algebra because simplified forms are easier to compare, combine, and use in further calculations. Adding radicals like 3√2 + 5√2 = 8√2 only works when the radicands (the parts left under the sign) are the same — you must simplify first to identify which terms can be combined. In geometry, radical expressions arise constantly: the diagonal of a unit square is √2, the height of an equilateral triangle with side a is (a√3)/2, and the golden ratio involves √5. Mastering radical simplification is therefore a prerequisite for algebra, trigonometry, and calculus.
Radical Calculator Examples
Click on any example to load it into the calculator.
| Input | Simplified Result | Method |
|---|---|---|
| √50 (radicand=50, index=2) | 5√2 | Factor 50 = 5² × 2. Extract √(5²) = 5 outside, leaving √2 inside. Decimal: ≈ 7.0711. |
| ∛54 (radicand=54, index=3) | 3∛2 | Factor 54 = 3³ × 2. Extract ∛(3³) = 3 outside, leaving ∛2 inside. Decimal: ≈ 3.7798. |
| √144 (radicand=144, index=2) | 12 | 144 = 12² is a perfect square, so √144 = 12 exactly. No radical remains. |
| ⁵√96 (radicand=96, index=5) | 2·⁵√3 | Factor 96 = 2⁵ × 3. Extract the perfect 5th power: ⁵√(2⁵) = 2 outside, ⁵√3 inside. |
How to Use the Radical Calculator
- Enter the radicand — the number inside the root sign — in the Radicand field. It must be a positive integer (even-index roots of negative numbers are complex).
- Enter the index — the degree of the root — in the Index field. Use 2 for square roots, 3 for cube roots, or any integer ≥ 2 for higher-order roots.
- Click Simplify Radical. The result shows the simplified form as a coefficient times the remaining radical, plus a decimal approximation.
- If the radicand is a perfect nth power, the result will be a whole number with no radical sign.
- Click Reset to clear the fields and simplify a different radical expression.
Radical Calculator FAQ
What does it mean to simplify a radical?
Simplifying a radical means rewriting it so that the radicand has no perfect nth power factors remaining under the radical sign. The simplified form consists of a coefficient (extracted from perfect factors) multiplied by a residual radical. For example, √72 simplifies to 6√2 because 72 = 36 × 2 and √36 = 6.
How do I simplify a square root by hand?
Find the largest perfect square that divides the radicand. Take its square root as the coefficient and leave the remaining factor under the root. For √200: 200 = 100 × 2 and √100 = 10, so √200 = 10√2. Alternatively, prime-factor the radicand and group pairs: 200 = 2³ × 5² = (2 × 5²) × 2 → 10√2.
Can I simplify a radical with a negative radicand?
For odd-index roots (cube root, 5th root, etc.) a negative radicand is valid: ∛-8 = -2 because (-2)³ = -8. For even-index roots (square root, 4th root, etc.) a negative radicand produces an imaginary (complex) number, which is outside the scope of this calculator. Enter positive numbers for even indices.
What is a perfect square and how does it help simplify radicals?
A perfect square is a number whose square root is an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, … Identifying perfect square factors of the radicand lets you extract whole numbers from under the radical. Similarly, perfect cubes (1, 8, 27, 64, 125, …) are used to simplify cube roots.
What is the product rule for radicals?
The product rule states that ⁿ√(a × b) = ⁿ√a × ⁿ√b for non-negative a, b. It is the core rule used in simplification: split the radicand into a perfect-power factor and a remainder, take the root of the perfect-power factor, and leave the remainder under the sign.
Why do I sometimes see radicals written with a coefficient like 3√5?
The notation 3√5 (or 3·√5) means three times the square root of five. The coefficient 3 has been extracted from under the radical sign because it came from a perfect square factor. This form is called the simplified radical form and makes it easy to add like radicals: 3√5 + 2√5 = 5√5, just like combining like terms in algebra.