Radical Calculator - Simplify Radical Expressions

What is a Radical Expression?

A radical expression is a mathematical notation that represents the root of a number. The most familiar example is the square root symbol, but radicals can represent cube roots, fourth roots, and roots of any degree. In algebra, a radical expression consists of the radical sign, the radicand (the value inside), and the index (the root degree). When we write \sqrt[n]{x}, we are asking: "What number, raised to the power n, gives x?"

Radical expressions appear throughout algebra, geometry, trigonometry, and calculus. Whenever you solve a quadratic equation, calculate a distance, or work with right triangles, you are likely to encounter radicals. Simplifying these expressions is a core skill in mathematics because it makes them easier to compare, combine, and use in further calculations. Our radical calculator automates this process, letting you focus on understanding the math rather than grinding through arithmetic.

How to Simplify Radicals Step by Step

Simplifying a radical means rewriting it so the radicand is as small as possible. The key technique is prime factorization: break the radicand into its prime factors, then pull out groups equal to the root degree.

Here is the general procedure. First, find the prime factorization of the radicand. Next, examine each prime factor's exponent. Divide that exponent by the root degree. The quotient tells you how many copies of that prime to move outside the radical, and the remainder tells you how many stay inside. Finally, multiply all the extracted primes together for the coefficient, and multiply all the remaining primes for the new radicand.

For example, consider \sqrt{72}. The prime factorization of 72 is 2^3 \times 3^2. For the square root (degree 2), divide each exponent by 2. The prime 2 has exponent 3, giving quotient 1 and remainder 1. The prime 3 has exponent 2, giving quotient 1 and remainder 0. The coefficient becomes 2^1 \times 3^1 = 6, and the remaining radicand is 2^1 = 2. Therefore \sqrt{72} = 6\sqrt{2}.

Simplifying Square Roots

Square root simplification is the most common type of radical simplification. You are looking for perfect square factors hidden inside the radicand. Perfect squares are numbers like 4, 9, 16, 25, 36, 49, 64, 81, and 100. When you spot one of these factors, pull its square root outside.

Consider a few examples. \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} because 25 is a perfect square. \sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2}. \sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}. Notice a pattern: many common radicals simplify to a coefficient times \sqrt{2} or \sqrt{3}. Recognizing these quickly is useful for mental math and algebra.

If the number under the radical is a perfect square itself, the radical vanishes entirely. For instance, \sqrt{144} = 12 and \sqrt{625} = 25. Our calculator detects perfect roots automatically and labels them clearly.

Simplifying Cube Roots and Higher Roots

For cube roots, the process is identical, but you group prime factors in threes instead of twos. A perfect cube factor (8, 27, 64, 125, 216, etc.) can be pulled outside the cube root sign.

Take \sqrt[3]{250}. The factorization of 250 is 2 \times 5^3. The prime 5 has exponent 3, which divides evenly by 3, so one 5 moves outside. The prime 2 has exponent 1, which stays inside. The result is 5\sqrt[3]{2}.

Fourth roots work similarly with groups of four. \sqrt[4]{162} factors as 2 \times 3^4. The group of four 3s exits the radical, giving 3\sqrt[4]{2}. The same logic extends to fifth roots, sixth roots, and beyond. While higher-degree roots are less common in everyday math, they appear in advanced algebra and number theory.

Simplifying Radicals with Fractions

When the radicand is a fraction, you can apply the radical to the numerator and denominator separately thanks to the quotient rule for radicals: \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}. Simplify each part independently, then combine. For example, \sqrt{\frac{50}{18}} = \frac{\sqrt{50}}{\sqrt{18}} = \frac{5\sqrt{2}}{3\sqrt{2}} = \frac{5}{3}.

Our calculator accepts fraction inputs directly. Type a value like 75/12 and it will simplify the radical of the fraction, showing each step for both numerator and denominator.

Radical Rules and Properties

Several algebraic rules govern how radicals behave. Knowing these rules allows you to combine, multiply, and divide radical expressions confidently.

Product Rule

The product rule states that \sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}. This is the foundation of simplification: it lets you split a radical into factors and simplify each one. It also works in reverse, allowing you to combine two radicals of the same degree into one.

Quotient Rule

The quotient rule says \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}. This property is essential for simplifying radicals of fractions and for rationalizing denominators. If a denominator contains a radical, you can often eliminate it by multiplying numerator and denominator by a suitable expression.

Power Rule

The connection between radicals and exponents is expressed by \sqrt[n]{a^m} = a^{m/n}. This rule lets you convert between radical notation and fractional exponent notation, which is often more convenient for algebraic manipulation.

Nesting Rule

Nested radicals combine by multiplying their indices: \sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}. For instance, the square root of the cube root of a number is the same as the sixth root of that number.

Difference Between Radical Calculator and Nth Root Calculator

These two tools serve different mathematical needs. An nth root calculator computes the numerical (decimal) value of a root. It answers "what is the cube root of 250?" with the approximation 6.2996. This is useful when you need a concrete number for measurements, engineering, or science.

A radical calculator, on the other hand, performs symbolic simplification. It answers the same question with the exact expression 5\sqrt[3]{2}. This is the form preferred in algebra, textbook problems, and any context where you need an exact answer rather than an approximation. Symbolic form also makes it easier to combine like radicals, rationalize denominators, and perform algebraic operations.

Our site offers both tools. Use the Nth Root Calculator when you need decimal precision and the Radical Calculator when you need exact simplified form. Many students find it helpful to check their work with both, verifying that the decimal approximation of the simplified form matches the direct computation.

Applications of Radical Simplification

Simplifying radicals is not just an academic exercise. It appears in many practical contexts across mathematics and related fields.

In algebra, simplification is necessary for combining like terms, solving equations, and working with conjugates. You cannot add \sqrt{12} and \sqrt{27} until you simplify them to 2\sqrt{3} and 3\sqrt{3}, at which point they combine to 5\sqrt{3}.

In geometry, many formulas produce radical expressions. The diagonal of a square with side s is s\sqrt{2}. The height of an equilateral triangle with side a is \frac{a\sqrt{3}}{2}. Keeping these in simplified radical form preserves exact values and avoids rounding errors that accumulate through multi-step calculations.

In trigonometry, the exact values of sine and cosine at standard angles are expressed as simplified radicals. For example, \sin(60°) = \frac{\sqrt{3}}{2} and \cos(45°) = \frac{\sqrt{2}}{2}. These expressions arise naturally from the geometry of the unit circle.

In standardized testing and coursework, answers are typically expected in simplified radical form. Leaving an answer as \sqrt{72} when the expected form is 6\sqrt{2} may result in lost marks, even though both expressions represent the same value.

Features of Our Radical Calculator

• Beautiful Math Rendering: All results are displayed using KaTeX, producing publication-quality mathematical notation directly in your browser.
• Step-by-Step Solutions: See the complete simplification process including prime factorization, exponent grouping, and the final result at each stage.
• Any Root Degree: Simplify square roots, cube roots, fourth roots, and roots of any positive integer degree.
• Fraction Support: Enter fractions directly (e.g., 3/4) and the calculator will simplify the radical of the fraction.
• Prime Factorization Display: View the complete prime factorization of your radicand, rendered in clear mathematical notation.
• Quick Root Selection: One-click buttons for the most common root degrees (square, cube, fourth) for fast switching.
• Copy to Clipboard: Export all results in formatted text with a single click for use in homework, reports, or messaging.
• Example Calculations: Pre-loaded examples for square roots, perfect squares, cube roots, and fourth roots help you learn proper usage.