Radical Calculator

Simplify any radical expression, including square roots, cube roots, and nth roots.

Enter a number (radicand) and the root index to find the simplified radical form.

Examples

Click on any example to load it into the calculator.

Simplify a Square Root

Square Root

Simplifying the square root of 50.

Radicand: 50

Index: 2

Simplify a Cube Root

Cube Root

Simplifying the cube root of 54.

Radicand: 54

Index: 3

Simplify a Perfect Square

Square Root

Simplifying the square root of a perfect square.

Radicand: 144

Index: 2

Simplify a Higher-Order Root

Nth Root

Simplifying the 5th root of 96.

Radicand: 96

Index: 5

Other Titles
Understanding Radicals: A Comprehensive Guide
Master the art of simplifying radicals, from basic square roots to complex nth roots, and understand their importance in mathematics.

What is a Radical? Core Concepts Explained

  • A radical expression involves a root, typically a square root.
  • Key components: radicand, index, and the radical symbol (√).
  • Simplifying a radical means extracting perfect powers from the radicand.
A radical is a mathematical expression that involves a root, denoted by the radical symbol (√). The number under the radical symbol is called the radicand, and the degree of the root is called the index. When no index is specified, it is assumed to be 2 (a square root).
The Anatomy of a Radical
An expression like 'ⁿ√b' consists of: 'n' as the index, 'b' as the radicand, and '√' as the radical symbol. The goal of simplification is to rewrite this expression in its simplest form, which is typically 'a * ⁿ√c', where 'c' has no perfect nth power factors.
For example, √50 is not in its simplest form because 50 contains a perfect square factor, 25 (since 5² = 25). We can rewrite √50 as √(25 2), which simplifies to √25 √2 = 5√2.

Basic Simplification Examples

  • √18 simplifies to 3√2 because 18 = 9 * 2 and √9 = 3.
  • ³√16 simplifies to 2³√2 because 16 = 8 * 2 and ³√8 = 2.
  • ⁴√80 simplifies to 2⁴√5 because 80 = 16 * 5 and ⁴√16 = 2.

Step-by-Step Guide to Using the Radical Calculator

  • Enter the radicand and index to begin.
  • Understand how the calculator processes the input.
  • Interpret the simplified output correctly.
Our Radical Calculator is designed for ease of use and accuracy. Follow these simple steps to simplify any radical expression.
Input Guidelines:
  • Radicand: Enter the number inside the radical symbol. It must be an integer. For even indices (like square roots), it cannot be negative.
  • Index: Enter the degree of the root. This must be an integer greater than or equal to 2. For a square root, enter 2.
Calculation and Results:
Once you click 'Simplify Radical', the calculator finds the largest nth power that divides the radicand. It then extracts this power, leaving the simplest possible number inside the radical.
The result is displayed in multiple parts: the simplified form (e.g., '5√2'), the coefficient (the number outside, e.g., 5), and the new radicand (the number inside, e.g., 2).

Practical Usage Examples

  • Input: Radicand = 72, Index = 2 -> Output: 6√2
  • Input: Radicand = 192, Index = 3 -> Output: 4³√3
  • Input: Radicand = 1, Index = 10 -> Output: 1

Real-World Applications of Radicals

  • Geometry: Calculating distances with the Pythagorean theorem.
  • Engineering: Used in electrical engineering for impedance calculations.
  • Finance: Calculating compound interest over fractional periods.
Radicals are not just an abstract algebraic concept; they appear frequently in various scientific, engineering, and financial fields.
Geometry and Physics
The most famous application is the Pythagorean theorem (a² + b² = c²), where c = √(a² + b²). This formula is fundamental in construction, navigation, and physics for calculating distances and magnitudes of vectors.
Electrical Engineering
In AC circuits, the concept of impedance involves complex numbers and radicals. The magnitude of impedance, which is crucial for circuit analysis, is often calculated using square roots.
Statistics and Data Analysis
Standard deviation, a key measure of data dispersion, is calculated by taking the square root of the variance. This helps statisticians understand the spread of data points.

Industry Applications

  • Finding the hypotenuse of a right triangle with sides 3 and 4: √(3²+4²) = √25 = 5.
  • Calculating the standard deviation of a dataset.
  • Determining the rate of return in financial models.

Common Misconceptions and Correct Methods

  • Understanding that √(a + b) ≠ √a + √b.
  • Correctly handling negative numbers under radicals.
  • The importance of the index in simplification.
Simplifying radicals can be tricky, and several common mistakes can lead to incorrect answers. Understanding these pitfalls is key to mastering the topic.
Mistake 1: Distributing the Root Over Addition
A frequent error is assuming that the root of a sum is the sum of the roots. For example, √(9 + 16) = √25 = 5. However, √9 + √16 = 3 + 4 = 7. It's crucial to remember that √(a + b) is not equal to √a + √b.
Mistake 2: Negative Radicands
The square root (or any even-indexed root) of a negative number is not a real number. For example, √-4 is undefined in the real number system. However, odd-indexed roots of negative numbers are real. For instance, ³√-8 = -2, because (-2)³ = -8.
Mistake 3: Ignoring the Index
When simplifying, you must look for factors that are perfect powers of the index. For ³√54, you look for perfect cubes (8, 27, 64...), not perfect squares. The correct factor is 27, leading to the simplified form 3³√2.

Avoiding Common Errors

  • Correct: √(100 - 36) = √64 = 8. Incorrect: √100 - √36 = 10 - 6 = 4.
  • ³√-27 = -3. But √-9 is not a real number.
  • To simplify ⁴√48, find a perfect 4th power factor: 16. So ⁴√48 = ⁴√(16 * 3) = 2⁴√3.

Mathematical Derivation and Formulas

  • The product property of radicals: ⁿ√(ab) = ⁿ√a * ⁿ√b.
  • The quotient property of radicals: ⁿ√(a/b) = ⁿ√a / ⁿ√b.
  • The process of prime factorization for simplification.
The simplification of radicals is based on a few key properties derived from the laws of exponents.
The Product Property
The most important rule for simplification is the product property: ⁿ√(ab) = ⁿ√a * ⁿ√b. This allows us to separate a radicand into factors and take the root of each factor individually. This is the foundation of the method used by the calculator.
The Simplification Algorithm
To simplify ⁿ√b, the algorithm is as follows:
1. Find the prime factorization of the radicand b.
2. For each prime factor, group them in sets of the index n.
3. For each complete group, move the prime factor outside the radical. This becomes the coefficient.
4. The prime factors that are left over remain inside the radical.
For example, to simplify √72: 1. Prime factorization of 72 is 2 2 2 3 3. 2. Index is 2, so we look for pairs: (22) 2 (33). 3. We have a pair of 2s and a pair of 3s. So we move one 2 and one 3 outside: 2 * 3 = 6. 4. One 2 is left inside. The result is 6√2.

Formulas in Action

  • √98 = √(49 * 2) = √49 * √2 = 7√2
  • ³√250 = ³√(125 * 2) = ³√125 * ³√2 = 5³√2
  • ⁴√162 = ⁴√(81 * 2) = ⁴√81 * ⁴√2 = 3⁴√2