What is a radical in math?

A radical is a mathematical expression that uses a root symbol (√). The most common radical is the square root (√), but radicals can be any root: cube root (∛), fourth root (∜), and so on. The number under the root sign is called the radicand, and the small number in the crook of the root sign is the index (2 for square root, 3 for cube root, etc.). For example, in ∛27, the radicand is 27 and the index is 3.

How do you simplify a radical?

To simplify a radical like √72, find the largest perfect square that divides evenly into the radicand. For √72: the largest perfect square factor is 36 (since 72 = 36 × 2). Then √72 = √(36 × 2) = √36 × √2 = 6√2. For cube roots, look for perfect cube factors instead. The key is prime factorization - break the number into primes, group them by the index, and pull out complete groups.

How do you multiply radicals?

To multiply radicals with the same index, multiply the coefficients together and multiply the radicands together, then simplify. For example: 3√5 × 2√10 = (3×2)√(5×10) = 6√50. Then simplify √50 = 5√2, so the final answer is 6 × 5√2 = 30√2. If the radicals have different indices, you need to convert them to the same index first using fractional exponents.

How do you add or subtract radicals?

You can only add or subtract radicals that have the same index and the same radicand (called like radicals). Just add or subtract the coefficients. For example: 3√5 + 7√5 = 10√5. If the radicands look different, try simplifying first - they might become like radicals. For instance, √12 + √27 = 2√3 + 3√3 = 5√3. If they cannot be made alike, the expression is already in simplest form.

How do you convert between radical and exponential form?

The rule is: the nth root of x to the m power equals x to the power m/n. Written as a formula: ⁿ√(xᵐ) = x^(m/n). So √x = x^(1/2), ∛x = x^(1/3), and ∛(x²) = x^(2/3). Going the other direction, x^(3/4) means the 4th root of x cubed: ∜(x³). This conversion is useful for calculus and when working with fractional exponents.

What is simplest radical form?

A radical is in simplest form when three conditions are met: (1) no perfect nth power factors remain under the radical sign, (2) the index is as small as possible, and (3) no radicals appear in the denominator (rationalized). For example, √50 is not simplified because 50 contains the perfect square factor 25. The simplified form is 5√2. Similarly, 1/√3 is not simplified until you rationalize it to √3/3.

Can you take the square root of a negative number?

Not in the real number system. The square root of a negative number is imaginary. For example, √(-9) = 3i, where i is the imaginary unit defined as √(-1). However, odd roots of negative numbers are real: ∛(-27) = -3 because (-3)³ = -27. Our calculator works with real numbers, so it will flag an error if you try an even root of a negative radicand.

What is the difference between a radical and an exponent?

Radicals and exponents are two notations for the same mathematical concept. A radical uses the root symbol (√), while an exponent uses a fractional power. They are interchangeable: √x = x^(1/2), ∛x = x^(1/3). Exponent form is often preferred in algebra and calculus because exponent rules (product rule, power rule, quotient rule) are easier to apply than radical rules. Our converter tool switches between both forms instantly.

Free Radical Calculator

Radical Calculator

Simplify any radical to its simplest form, perform operations on radical expressions, or convert between radical and exponential notation. Step-by-step solutions for every calculation.

Calculator Mode

Simplify a Radical

Coefficient

Index (n)

Radicand (number under the root)

Quick examples:

Result

Enter values to calculate

Perfect Squares

1²1

2²4

3²9

4²16

5²25

6²36

7²49

8²64

9²81

10²100

11²121

12²144

13²169

14²196

15²225

Perfect Cubes

1³1

2³8

3³27

4³64

5³125

6³216

7³343

8³512

9³729

10³1000

Common Radical Simplifications

Quick reference for frequently simplified square roots.

RadicalFactorizationSimplified

√8√(4×2)2√2

√12√(4×3)2√3

√18√(9×2)3√2

√20√(4×5)2√5

√45√(9×5)3√5

√48√(16×3)4√3

√50√(25×2)5√2

√72√(36×2)6√2

√75√(25×3)5√3

√98√(49×2)7√2

√128√(64×2)8√2

√200√(100×2)10√2

How to Use the Radical Calculator

Simplify Mode

Enter a radicand (the number under the root) and choose the index (square root, cube root, 4th root, or 5th root). Optionally add a coefficient in front. The calculator factors the radicand, pulls out perfect powers, and shows every step of the simplification.

Operations Mode

Enter two radical expressions with coefficients and radicands, pick an operation (add, subtract, multiply, or divide), and see the result with full working. Addition and subtraction require like radicands after simplification. Multiplication and division always produce a result.

Convert Mode

Switch between radical notation (√, ∛, ∜) and exponential notation (x^(m/n)). Radical-to-exponential is useful for calculus. Exponential-to-radical helps when you need to visualize what a fractional exponent actually means.

Why Use This Radical Calculator

Step-by-Step Work

Every simplification shows the factorization, the extraction of perfect powers, and the final result. Follow the logic instead of just getting an answer.

Three Tools in One

Simplify, operate, and convert - all on the same page. No need to switch between different calculators for different radical tasks.

Any Root Index

Square roots, cube roots, fourth roots, fifth roots. Enter any index from 2 upward and the calculator adapts its factorization accordingly.

Perfect Power Reference

Side panels list perfect squares (1 through 225) and perfect cubes (1 through 1000) so you can spot factors at a glance while working problems.

Decimal Approximation

Every result includes a decimal approximation alongside the exact radical form. Useful when you need a numerical value for applied problems or checking your work.

Export Solutions

Copy the result to clipboard or download a text file with the full step-by-step solution. Attach it to homework, paste into study notes, or share with classmates.

A Complete Guide to Radicals

What Radicals Are and Where They Come From

The word "radical" comes from the Latin radix, meaning root. In mathematics, a radical expression involves taking a root of a number or expression. The square root is the most common: √25 asks "what number multiplied by itself gives 25?" The answer is 5. Cube roots ask the same question for three factors: ∛125 = 5 because 5 × 5 × 5 = 125.

Radicals have been part of mathematics since ancient Babylonian algebra, roughly 2000 BCE. The modern radical symbol (√) evolved from a stylized letter "r" for radix in the 1500s. Today radicals appear throughout algebra, geometry (the Pythagorean theorem produces square roots constantly), trigonometry, physics, engineering, and anywhere measurements involve diagonal distances or areas.

The Art of Simplifying Radicals

Simplifying a radical means rewriting it so that no perfect power factor remains under the root sign. The method relies on one key property: ⁿ√(a × b) = ⁿ√a × ⁿ√b. You exploit this by splitting the radicand into a perfect-power factor and a leftover.

For square roots, you look for the largest perfect square factor. Take √180: prime factorize to get 2² × 3² × 5. Pull out 2 and 3 (the complete pairs), leaving 5 inside. Result: 6√5. For cube roots, you need groups of three: ∛216 = ∛(6³) = 6, a perfect cube. ∛250 = ∛(125 × 2) = 5∛2.

A systematic approach is prime factorization. Write the radicand as a product of primes, then for index n, pull out every group of n identical primes. Each group contributes one factor to the coefficient. Whatever primes remain ungrouped stay under the radical. This method never fails, even for large numbers.

Operations with Radical Expressions

Multiplying and dividing radicals is straightforward because the product and quotient rules let you combine radicands: √a × √b = √(ab) and √a / √b = √(a/b). After combining, simplify the result.

Addition and subtraction are more restrictive. You can only combine like radicals - expressions with the same index and the same simplified radicand. 4√3 + 2√3 = 6√3, just as 4x + 2x = 6x. But 4√3 + 2√5 stays as is because √3 and √5 are not like terms. The trick is to simplify each radical first: √12 + √27 looks uncombineable until you realize √12 = 2√3 and √27 = 3√3, giving 5√3.

Division sometimes requires rationalizing the denominator - multiplying top and bottom by the radical to eliminate it from the denominator. For 5/√3, multiply by √3/√3 to get 5√3/3. For a binomial denominator like 1/(√5 + √2), multiply by the conjugate (√5 - √2)/(√5 - √2). Our operations mode handles these calculations automatically.

Radicals and Exponents: Two Sides of One Coin

Every radical can be written as a fractional exponent: ⁿ√(xᵐ) = x^(m/n). This equivalence is not just notational convenience - it means all exponent rules apply directly. Want to multiply √x × ∛x? Convert: x^(1/2) × x^(1/3) = x^(1/2 + 1/3) = x^(5/6) = ⁶√(x⁵). Doing that with radical notation alone is clunky, but exponent form makes it a simple fraction addition.

Calculus relies heavily on this conversion. The derivative of √x is the derivative of x^(1/2), which is (1/2)x^(-1/2) = 1/(2√x) by the power rule. Without the exponent translation, you would need a separate differentiation rule for each type of root. The same applies to integration, where fractional exponents simplify antiderivatives enormously.

Where You Will See Radicals in Practice

The Pythagorean theorem (a² + b² = c²) is the most common source of radicals in geometry. A right triangle with legs 1 and 1 has a hypotenuse of √2. Legs of 1 and 2 give √5. These irrational values cannot be expressed as neat fractions, so they stay in radical form for exact work.

The quadratic formula, x = (-b ± √(b²-4ac)) / 2a, generates radicals whenever the discriminant is not a perfect square. Physics uses radicals in formulas for pendulum period (T = 2π√(L/g)), escape velocity (v = √(2GM/r)), and wave speed. Finance uses them in compound interest and volatility calculations. Even in everyday life, the distance formula between two GPS coordinates involves a square root.

Understanding how to simplify and manipulate radicals is not an abstract exercise - it is a skill that unlocks cleaner, more efficient problem-solving across dozens of fields.