Radical Calculator - Simplify Nth Roots Step by Step
Enter a coefficient, root index, and radicand to simplify any radical expression into its simplest form. The calculator performs prime factorization, extracts perfect nth-power factors outside the radical, and shows every step of the work. Handles square roots, cube roots, fourth roots, and any higher-order nth root, including negative radicands for odd indices.
What is a radical?
A radical (or root) is the inverse of raising a number to a power. The most common radical is the square root, written as sqrt(x), which asks: "what number, squared, gives x?" More generally, the nth root of x, written as nth-root(x) or x^(1/n), asks: "what number, raised to the nth power, gives x?" For example, the cube root of 27 is 3, because 3^3 = 27. The number inside the radical sign is called the radicand, and the small number indicating which root to take (2, 3, 4, etc.) is called the index. A coefficient multiplied in front of the radical scales the entire expression: 2*sqrt(3) means twice the square root of 3.
How to simplify a radical expression
Simplifying a radical means rewriting it so that the radicand has no perfect nth-power factors. The process uses prime factorization. First, express the radicand as a product of its prime factors. Then group those factors into sets of n (the root index). Each complete group of n identical factors can be moved outside the radical as a single factor, while any leftovers stay inside. For example, to simplify sqrt(72): 72 = 2^3 * 3^2. For the square root (n=2), group pairs: two 2s make one 2 outside, one 2 is left over, and two 3s make one 3 outside. Result: 2 * 3 * sqrt(2) = 6*sqrt(2). The coefficient out front multiplies by any factor extracted from the radical.
Negative radicands and complex numbers
Whether a negative radicand has a real solution depends on the root index. For even-index roots (square root, fourth root, sixth root, etc.), a negative radicand has no real solution because no real number raised to an even power gives a negative result. The result is an imaginary number using the unit i, where i = sqrt(-1). For odd-index roots (cube root, fifth root, seventh root, etc.), a negative radicand does have a real solution: the nth root of -x equals -(nth root of x). For example, the cube root of -8 is -2, because (-2)^3 = -8. This calculator computes real solutions only.
Rational exponents and the connection to roots
Roots and exponents are two sides of the same operation. The nth root of x is exactly equal to x raised to the power 1/n. More generally, x^(m/n) means the nth root of x, raised to the m power. This connection makes it easy to work with radicals using exponent rules: multiplying two radicals with the same index is equivalent to multiplying their radicands under one radical sign, and dividing works the same way. For example, sqrt(3) * sqrt(12) = sqrt(36) = 6. These rules only apply when the index is the same for both radicals.
Common radical simplification examples
| Expression | Simplified form | Decimal value | Notes |
|---|---|---|---|
| sqrt(4) | 2 | 2.000 | Perfect square |
| sqrt(8) | 2*sqrt(2) | 2.828 | 8 = 4 * 2, factor of 2 outside |
| sqrt(12) | 2*sqrt(3) | 3.464 | 12 = 4 * 3 |
| sqrt(18) | 3*sqrt(2) | 4.243 | 18 = 9 * 2 |
| sqrt(50) | 5*sqrt(2) | 7.071 | 50 = 25 * 2 |
| sqrt(72) | 6*sqrt(2) | 8.485 | 72 = 36 * 2 |
| cbrt(8) | 2 | 2.000 | Perfect cube |
| cbrt(24) | 2*cbrt(3) | 2.884 | 24 = 8 * 3 |
| cbrt(54) | 3*cbrt(2) | 3.780 | 54 = 27 * 2 |
| 4th-root(16) | 2 | 2.000 | Perfect fourth power |
| 4th-root(48) | 2*4th-root(3) | 2.632 | 48 = 16 * 3 |
Frequently used perfect powers and their nth roots, showing the simplification pattern.
Frequently asked questions
What does "simplest radical form" mean?
A radical is in its simplest form when the radicand contains no perfect nth-power factors (other than 1), the index is as small as possible, and there are no fractions under the radical sign. For a square root, this means no perfect square factors remain inside: sqrt(12) simplifies to 2*sqrt(3) because 4 is a perfect square factor of 12.
How do I know if a square root is a whole number?
A square root of a positive integer is a whole number only if that integer is a perfect square: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on. If the prime factorization of the radicand consists entirely of even-exponent primes, the square root is exact. This calculator marks the result "exact" when the radicand simplifies to 1 after extraction.
Can I simplify the cube root of a negative number?
Yes. Odd-index roots (cube root, fifth root, etc.) of negative numbers are real and negative. The cube root of -8 is -2 because (-2)^3 = -8. Enter the negative radicand and select index 3 (or any odd index). The calculator will compute and simplify the result, applying the negative sign to the final answer.
How do I add or subtract radicals?
Radicals can only be added or subtracted when they are "like radicals," meaning they have the same index and the same radicand after simplification. For example, sqrt(50) + sqrt(8) = 5*sqrt(2) + 2*sqrt(2) = 7*sqrt(2). Use this calculator to simplify each radical first, then check whether the radicands match. If they do not match after simplification, the radicals cannot be combined into a single term.
What is the difference between the square root and the principal square root?
Squaring either 3 or -3 gives 9, so technically 9 has two square roots: 3 and -3. The principal square root is defined as the non-negative one (3), which is what the sqrt symbol conventionally means and what this calculator returns. If you need both roots (as in solving x^2 = 9), remember that the solutions are +sqrt(9) and -sqrt(9).
Why does the decimal approximation sometimes look imprecise?
Most irrational radical values (like sqrt(2) = 1.41421356...) are non-terminating decimals that cannot be stored exactly in a computer. The calculator displays up to 8 decimal places, which is sufficient for nearly all practical uses. For exact answers, use the simplified radical form rather than the decimal approximation.