Radical Calculator

Find the largest perfect cube factor of the radicand. For example, ∛54 = ∛(27 × 2) = 3∛2 because 27 is the largest perfect cube dividing 54.

A perfect square is a number whose square root is an integer (e.g., 25 = 5²). A radical is an irrational root that cannot be simplified to a whole number (e.g., √2).

You can only add or subtract radicals with the same radicand and index. For example, 3√2 + 5√2 = 8√2, but you cannot directly add √2 + √3.

Multiply both numerator and denominator by the radical to eliminate the root from the denominator. For example, 1/√2 becomes √2/2 after rationalizing.

The nth root of a number a is a value that, when raised to the nth power, equals a. For example, ∛8 = 2 because 2³ = 8.

The principal square root is the non-negative square root. For example, √9 = 3 (not -3), even though both 3 and -3 squared give 9.

Multiply the radicands and keep the same index: √a × √b = √(a×b). For example, √3 × √12 = √36 = 6.

Divide the radicands and keep the same index: √a / √b = √(a/b). Rationalize the denominator if needed.

√(a×b) = √a × √b. This rule allows you to break apart or combine radicals under certain conditions.

√(a/b) = √a / √b. This rule allows you to separate the numerator and denominator under a radical.

50 = 25 × 2 = 5² × 2, so √50 = √(25 × 2) = 5√2.

24 = 8 × 3 = 2³ × 3, so ∛24 = ∛(8 × 3) = 2∛3.

The conjugate of a + √b is a - √b. Multiplying a radical expression by its conjugate eliminates the radical: (a + √b)(a - √b) = a² - b.

Multiply both numerator and denominator by the conjugate of the denominator. For example, 1/(1 + √2) becomes (1 - √2)/(-1) = √2 - 1.

Like radicals have the same index and same radicand. For example, 3√5 and 7√5 are like radicals and can be combined: 3√5 + 7√5 = 10√5.

Find the two perfect squares (or cubes) the radicand falls between. For example, √50 is between √49 = 7 and √64 = 8, so it's approximately 7.1.

The radical symbol √ was first used by Christoph Rudolff in 1525. Before that, various words and symbols were used to denote roots.

The Pythagorean theorem c = √(a² + b²) uses a square root to find the hypotenuse of a right triangle. Simplifying the radical gives the exact answer.

Radicals can be written as fractional exponents: √a = a^(1/2), ∛a = a^(1/3), etc. This connection allows using exponent rules with radicals.

√[n]{a} = a^(1/n). For example, √x = x^(1/2) and ∛x = x^(1/3).

Even-indexed roots (square, fourth) of negative numbers are not real. Odd-indexed roots (cube, fifth) of negative numbers are negative real numbers.

Nested radicals like √(2 + √3) can sometimes be simplified by assuming √(a + √b) = √x + √y and solving for x and y.

A radical is in simplest form when: (1) no perfect nth power divides the radicand, (2) no fractions are under the radical, and (3) no radicals are in the denominator.

Convert to a common index (the LCM of the indices), then combine. For example, √2 + ∛2 requires converting to a common index before adding.

Square roots of negative numbers are imaginary: √(-1) = i. This extends the radical concept to the complex number system.

Radicals appear in derivatives (power rule with fractional exponents), integrals, and limits. Simplifying radicals often makes calculus computations easier.

The Radical Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Radical Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.