Radical Simplifier
Square roots · cube roots · nth roots · rationalize denominator · radical expressions
Enter a positive integer to simplify its square root by factoring out the largest perfect square.
Simplify any nth root by factoring out the largest perfect nth-power. Example: ∛54 → 3∛2
Perform operations on radical expressions of the form a√b ± c√d (add/subtract) or a√b × c√d (multiply) or a√b ÷ c√d (divide).
Rationalize the denominator of a fraction. Choose between a monomial denominator (p / (c√d)) or a binomial denominator (p / (a√b ± c√d)).
Expression: p / (c√d)
Expression: p / (a√b ± c√d)
Convert between radical notation and fractional exponent form. Example: ∛x² = x^(2/3).
Expression: ⁿ√(x^m) = x^(m/n)
Expression: x^(m/n) = ⁿ√(x^m)
What Are Radicals and Why Simplify Them?
A radical is a mathematical expression that involves a root — most commonly a square root (√), cube root (∛), or an nth root. The symbol √ is called a radical sign, the number inside is the radicand, and the small number written above the radical sign (when present) is the index. For a square root, the index is 2 and is typically omitted by convention.
Simplifying radicals means rewriting a radical expression in its most reduced form so that the radicand has no perfect-power factors. For instance, √72 contains a hidden factor of 36 (a perfect square), so it simplifies to √(36 × 2) = 6√2. The number 6√2 is the simplified form because 2 has no perfect-square factors.
You need to simplify radicals in virtually every area of mathematics: solving quadratic equations, computing distances with the Pythagorean theorem, simplifying trigonometric expressions, rationalizing fractions in calculus, and working with irrational numbers in algebra. Engineers use simplified radical forms when computing resonant frequencies, signal amplitudes, and structural load calculations. Simplified radicals are also far easier to combine with other like radicals — just as you would group like terms in algebra.
How to Use This Calculator
- Choose a mode from the tabs: Simplify √n, nth Root, Radical Expressions, Rationalize, or Radical ↔ Exponent.
- Enter your values in the input fields. All fields accept integers; decimal inputs are rounded where necessary.
- Click the action button (Simplify / Calculate / Rationalize / Convert) or press a quick-example button to pre-fill a sample.
- Read the step-by-step solution that appears below the inputs. Each step is numbered and explains the rule applied.
- The final result is highlighted in the summary card along with a decimal approximation.
Key Simplification Rules
| Rule | Formula | Example |
|---|---|---|
| Product rule | √(ab) = √a × √b | √12 = √4 × √3 = 2√3 |
| Quotient rule | √(a/b) = √a / √b | √(9/4) = 3/2 |
| Perfect square factor | √(k²·m) = k√m | √72 = √(36·2) = 6√2 |
| nth root product | ⁿ√(ab) = ⁿ√a × ⁿ√b | ∛54 = ∛(27·2) = 3∛2 |
| Radical exponent | ⁿ√(xᵐ) = x^(m/n) | ∛x² = x^(2/3) |
| Adding like radicals | a√k + b√k = (a+b)√k | 3√5 + 7√5 = 10√5 |
| Multiplying radicals | a√b × c√d = ac√(bd) | 2√3 × 4√3 = 8×3 = 24 |
| Rationalize (mono) | p/√d = p√d/d | 1/√2 = √2/2 |
| Rationalize (binom) | p/(√a+√b) = p(√a−√b)/(a−b) | 1/(√3+√2) = √3−√2 |
Worked Examples
Example 1 — Simplify √72
Find the largest perfect square that divides 72. The divisors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The perfect squares among these are 1, 4, 9, 36. The largest is 36.
√72 = √(36 × 2) = √36 × √2 = 6√2
Decimal approximation: 6 × 1.41421… ≈ 8.48528
Example 2 — Simplify ∛54
Find the largest perfect cube that divides 54. Cubes: 1, 8, 27, 64… Divisors of 54: 1, 2, 3, 6, 9, 18, 27, 54. Largest perfect cube factor: 27.
∛54 = ∛(27 × 2) = ∛27 × ∛2 = 3∛2
Decimal approximation: 3 × 1.25992… ≈ 3.77976
Example 3 — Add 2√3 + 5√3
Both terms have the same radicand (3) and the same index (2). They are like radicals and can be combined by adding their coefficients:
2√3 + 5√3 = (2 + 5)√3 = 7√3
Example 4 — Rationalize 1/√5
Multiply numerator and denominator by √5:
1/√5 × (√5/√5) = √5 / (√5 × √5) = √5 / 5 = √5/5
The denominator is now rational (no radical sign). Decimal: ≈ 0.44721
Frequently Asked Questions
Why do we simplify radicals?
What is rationalization of the denominator?
1/√5, multiply numerator and denominator by √5 to get √5/5. For a binomial denominator like 1/(√3+√2), multiply by the conjugate (√3−√2) to exploit the difference-of-squares identity and eliminate both radicals at once.