Radical Calculator
Our free arithmetic calculator solves radical problems. Get worked examples, visual aids, and downloadable results. Get results you can export or share.
Formula
nth_root(x) = x^(1/n); sqrt(a*b) = sqrt(a) * sqrt(b)
The nth root of x equals x raised to the power 1/n. Radicals can be simplified by extracting perfect nth power factors. Product and quotient rules allow multiplication and division of same-index radicals by combining radicands.
Worked Examples
Example 1: Simplifying the Square Root of 72
Problem: Simplify the square root of 72 and express in simplest radical form.
Solution: Prime factorize 72: 72 = 2^3 * 3^2\nGroup in pairs for square root:\n2^3 = 2^2 * 2 (one pair of 2s extracted, one 2 remains)\n3^2 = one pair of 3s extracted\n\nExtracted: 2 * 3 = 6\nRemaining under radical: 2\n\nResult: 6 * sqrt(2)\nDecimal: 6 * 1.41421 = 8.48528\nVerification: (6 * sqrt(2))^2 = 36 * 2 = 72
Result: sqrt(72) = 6*sqrt(2) = 8.48528
Example 2: Multiplying Cube Roots
Problem: Multiply the cube root of 12 by the cube root of 18 and simplify.
Solution: cube_root(12) * cube_root(18) = cube_root(12 * 18) = cube_root(216)\n\n216 = 6^3, so cube_root(216) = 6\n\nAlternatively:\n12 = 2^2 * 3\n18 = 2 * 3^2\n12 * 18 = 2^3 * 3^3 = (2*3)^3 = 6^3\n\nResult: 6 (a perfect cube!)
Result: cube_root(12) * cube_root(18) = cube_root(216) = 6
Frequently Asked Questions
What is a radical in mathematics and how do you read radical notation?
A radical is a mathematical expression that represents the root of a number. The radical symbol consists of a check mark shape with a horizontal bar extending over the radicand (the number under the radical). The small number in the notch of the radical symbol is the index, indicating which root to take. A square root (index 2) is the most common and typically written without showing the index. A cube root has index 3, a fourth root has index 4, and so on. The expression denoting the nth root of x asks: what number raised to the nth power equals x? For example, the cube root of 27 is 3 because 3 cubed equals 27. Radicals are inverse operations to exponentiation, just as subtraction is the inverse of addition and division is the inverse of multiplication.
Can I use Radical Calculator on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.
Is Radical Calculator free to use?
Yes, completely free with no sign-up required. All calculators on NovaCalculator are free to use without registration, subscription, or payment.
How accurate are the results from Radical Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.