What is a radical in mathematics?

A radical represents an nth root of a number. The number under the radical is called the radicand.

How do you simplify a radical?

Use prime factorization, group factors by the index, and pull complete groups outside the radical. Example: sqrt(72) = 6 sqrt(2).

What is the difference between a square root and cube root?

A square root asks for a number squared to get the radicand, while a cube root asks for a number cubed.

Can radicals be negative?

Even roots of negative numbers are complex. Odd roots of negative numbers are real, such as cube root of -8 equals -2.

When should I rationalize a denominator with radicals?

Rationalize when a radical is in the denominator and your class or exam expects simplified form. Multiply by an appropriate radical factor or conjugate so the denominator becomes rational.

Radical Calculator - Simplify Nth Roots with Step-by-Step Solutions

Radical Calculator

Simplify square roots, cube roots, and nth roots with step-by-step solutions using prime factorization and simplification methods. Master radical expressions with our educational tool.

100% FreeStep-by-step SolutionsLearning-focused

Radical Simplifier

Enter the values for your radical expression: y · √[n](x)

Enter to calculate, Esc to clear

Coefficient (y)

Default: 1

Root Index (n)

2 = square root, 3 = cube root, etc.

Radicand (x) *

Number under the radical sign

Common Radical Examples

Try these popular radical simplification examples

\sqrt[2]{72}

= $6\sqrt{2}$

\sqrt[2]{50}

= $5\sqrt{2}$

\sqrt[3]{24}

= $2\sqrt[3]{3}$

2\sqrt[2]{18}

= $6\sqrt{2}$

\sqrt[2]{128}

= $8\sqrt{2}$

What is a Radical?

A radical is a mathematical expression that uses a root symbol (√) to indicate the nth root of a number. The general form is $\sqrt[n]{x}$ , where:

n is the root index (2 for square root, 3 for cube root, etc.)
x is the radicand (the number under the radical sign)
When n = 2, we write $\sqrt{x}$ (square root)
When n = 3, we write $\sqrt[3]{x}$ (cube root)

Example: $\sqrt[5]{32} = 2$ because $2^5 = 32$

How to Simplify Radicals?

Method 1: Prime Factorization

Find the prime factorization of the radicand
Group factors into sets of n (root index)
Extract perfect nth powers from under the radical
Multiply by any coefficient outside

Method 2: Step-by-Step Simplification

Identify perfect nth power factors
Extract these factors from the radical
Combine with the coefficient
Leave remaining factors under the radical

Why Use Our Radical Calculator?

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Learn the process with detailed explanations

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How √2 Makes Your Paper Perfect: The Secret Behind A4 Sizes

Ever wonder why A4 paper has such an odd size (210 × 297 mm)? The answer is $\sqrt{2}$ — and it's one of the most elegant applications of radicals in everyday life.

The Magic Ratio: 1 : √2

Every sheet in the ISO 216 standard (A0, A1, A2, A3, A4...) has the ratio:

\frac{\text{long side}}{\text{short side}} = \sqrt{2} \approx 1.4142

Why this specific ratio? When you fold it in half, the new sheet has the exact same proportions:

\frac{w}{h/2} = \frac{w}{h/2} = \frac{2w}{h} = \frac{2}{\sqrt{2}} = \sqrt{2}

No other ratio has this self-similar folding property. It's mathematically unique.

📐 TV Diagonal Math

A "55-inch TV" means 55 inches diagonally. For a 16:9 TV, the actual width and height are:

w = \frac{55 \times 16}{\sqrt{16^2 + 9^2}} = \frac{880}{\sqrt{337}} \approx 47.9''

The radical in the denominator comes from the Pythagorean theorem!

🔊 Audio: The -3dB Point

In audio engineering, $\sqrt{2}$ appears as the "half-power point." When a signal drops by $\frac{1}{\sqrt{2}}$ in voltage, power halves — that's exactly -3 dB.

P = \frac{V^2}{R} \rightarrow \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}

📖 ISO 216: Paper Sizes Standard 📘 OpenStax Intermediate Algebra 2e

Frequently Asked Questions

A radical expression contains a root symbol (√). The number under the radical is the radicand, and the small number (index) indicates which root to take. For example, ∛8 means the cube root of 8, which equals 2.

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