Lesson 1.10
Simplifying Radicals
Square roots aren't always nice integer numbers. We learn to express numbers like in their simplest exact form, a critical skill for the Quadratic Formula.
Introduction
Just as we reduce fractions like to , we must simplify square roots. We extract "perfect square" factors to make the number inside the radical as small as possible.
Past Knowledge
You know perfect squares: .
Today's Goal
Write irrational numbers exactly. Instead of , we write .
Future Success
This is mandatory for the Quadratic Formula (Chapter 4) and Trigonometry.
Key Concepts
Product Property of Radicals
This allows us to break a number apart. We look for a Perfect Square factor.
The "Jailbreak" Analogy
Think of the square root symbol as a jail.
- Numbers can only escape in pairs.
- A pair of twins merges into one person outside.
- Remainders stay inside.
Example: for , factors are . The pair of 3s escapes as a single 3.
Perfect Squares List
Memorize these! They are your "keys" to unlock the radical.
Worked Examples
Example 1: Basic Simplification
BasicSimplify .
Find Largest Perfect Square Factor
Factors of 18: 1, 2, 3, 6, 9, 18.
Perfect squares in that list: 1, 9.
The largest is 9.
Split and Simplify
Result:
Example 2: With a Coefficient
IntermediateSimplify .
Find Largest Square in 32
Don't use 4! Use 16. ().
Extract and Multiply
Result:
Example 3: Fraction (Quotient Property)
AdvancedSimplify .
Split Top and Bottom
Simplify Each
Result:
Common Pitfalls
Stopping too soon
For , finding is incomplete because 8 still has a factor of 4. Always check if the inside number (radicand) has more perfect square factors.
Adding Radicals Incorrectly
, NOT . You treat radicals like variables ().
Real-Life Applications
Exact Distance (GPS)
In navigation and game design, distances are calculated using the Pythagorean Theorem ().
If you walk 2 miles East and 4 miles North, the distance is . Simplifying to helps estimations and keeps calculations precise.
Practice Quiz
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