Enter a radicand (the number under the radical) and an index (the root degree) to evaluate any radical expression. The Radical Calculator computes the exact decimal result, simplifies the expression in surd form where possible, and shows the equivalent exponent notation — all in one step.
Decimal Result
--
Simplified Radical Form
--
Exponent Notation
--
Simplified: Outside Factor
--
Simplified: Remaining Radicand
--
A radical expression contains a radical symbol (√) with a radicand — the number or expression under the symbol — and an index indicating which root to take. For example, ⁴√81 means the 4th root of 81, which equals 3. When no index is written, a square root (index = 2) is assumed.
Radicals and exponents are directly connected: the n-th root of x equals x raised to the power of 1/n. So √x = x^(1/2), ∛x = x^(1/3), and ⁿ√x = x^(1/n). This relationship allows radical expressions to be rewritten as fractional exponents, which is often easier for algebraic manipulation.
To simplify a radical, find the largest perfect n-th power factor of the radicand. Extract that factor outside the radical, and leave the remaining factor inside. For example, √72 = √(36 × 2) = 6√2, because 36 is a perfect square. The calculator handles this factoring automatically.
For even-index roots (square root, 4th root, etc.), the radicand must be non-negative when working with real numbers — negative inputs produce no real result. For odd-index roots (cube root, 5th root, etc.), negative radicands are valid and produce negative results. For example, ∛(−8) = −2.
A coefficient multiplies the entire radical result. For example, 3√72 means 3 times the square root of 72. In simplified form, 3√72 = 3 × 6√2 = 18√2. You can enter any coefficient in the calculator to account for expressions like 2√50 or 5∛27.
Simplified radicals are easier to compare, add, and use in further calculations. For instance, adding √8 and √2 is unclear until you simplify √8 to 2√2, after which the sum becomes 3√2. Simplified forms are also standard in academic and professional contexts.
Common errors include assuming √(a + b) = √a + √b (this is false), forgetting to check for extraneous solutions when solving radical equations, and incorrectly applying the index when roots other than square roots are involved. Always verify your answer by substituting it back into the original expression.
Radicals appear in geometry (e.g., the Pythagorean theorem gives side lengths as square roots), physics (e.g., calculating velocity or wave frequency), engineering (e.g., stress and signal calculations), and finance (e.g., compound interest formulas and standard deviation in statistics).