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Reduced Radical Form

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Reduced Radical Form

Study Guide

Key Definition

An expression is in reduced radical form if the radicand has no perfect square factors other than 1. For example, $\sqrt{18} = 3\sqrt{2}$.

Important Notes

A radicand is the number under the radical sign.
Perfect squares are numbers like 4, 9, and 16, which are squares of integers.
To simplify $\sqrt{50}$, factor out the perfect square: $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$.
The product property of square roots helps in simplification: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$.
Cubic roots use similar logic but with perfect cubes.

Mathematical Notation

$\sqrt{}$ denotes a square root

$\times$ represents multiplication

$\div$ represents division

$x^2$ denotes x squared

Why It Works

By factoring the radicand into its prime factors, you can identify and extract perfect squares, thus simplifying the expression. This uses the property $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$.

Remember

Always factor the radicand completely to find and simplify using perfect squares: $\sqrt{36} = 6$.

Quick Reference

Product Property:$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

Understanding Reduced Radical Form

Choose your learning level

Beginner

Start here! Easy to understand

Beginner Explanation

Simple explanation with $\sqrt{4} = 2$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the reduced form of $\sqrt{12}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You need to find the length of a diagonal of a square with a side of $\sqrt{8}$ meters.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

What is the reduced form of $\sqrt{72}$?

Challenge Quiz

Single Choice Quiz

Advanced

Which of these is the reduced form of $\sqrt{200}$?

Recap

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