Simplify Radical Expressions Calculator

What is a Radical Expression?

A radical expression is like a special math puzzle! It includes a square root symbol (√) or other root symbols. These expressions help us work with numbers that, when multiplied by themselves a certain number of times, give us another number.

How to Simplify Radical Expressions

Simplifying radical expressions is like organizing a messy room. We look for ways to make the expression neater and easier to understand. We do this by finding factors that can be taken out of the radical.

Formula

We write a radical expression like this:

\[ y \cdot \sqrt[n]{x} \]

This means:

• \(y\) is the number outside the radical (like a regular number)
• \(n\) is the index (what root we're taking)
• \(x\) is the radicand (the number under the radical sign)
• \(\sqrt[n]{}\) is the radical symbol

Calculation Steps

1. Look at the number under the radical (x)
2. Find factors of x that are perfect nth powers
3. Take out these perfect nth power factors
4. Write the taken out factors outside the radical
5. Leave any remaining factors under the radical
6. Multiply any numbers outside the radical

Example and Visual Representation

Let's simplify this expression: \(3 \cdot \sqrt[3]{72}\)

Step 1: Look at 72 under the cube root

Step 2: Find factors: 72 = 2 × 2 × 2 × 3 × 3

Step 3: Take out \(2^3\) (a perfect cube)

Step 4: Write 2 outside and leave 3 × 3 inside

Step 5: Simplify: \(3 \cdot 2 \cdot \sqrt[3]{9}\)

Step 6: Multiply outside: \(6 \cdot \sqrt[3]{9}\)

In this picture, we show how the original expression changes to the simplified form. The arrow represents the simplification process we followed in our steps.