Lesson 8.2

The Quotient Rule

Multiplication gathered them up. Division cancels them out. It's the war of the exponents.

Introduction

In Lesson 8.1, adding matching bases meant adding exponents. Now we do the opposite. When we divide matching bases, we subtract the exponents.

Past Knowledge

Lesson 8.1 (Product Rule). You know that exponents count how many factors there are. Division removes factors.

Today's Goal

Simplify expressions like by subtracting exponents.

Future Success

This leads directly to Zero Exponents (Lesson 8.4) and Negative Exponents.

Key Concepts

The Rule

If the bases are the same, SUBTRACT the exponents (Top - Bottom).

Why?

Worked Examples

Example 1: Basic Subtraction

Basic

Simplify:

1

Check Bases

Both are . Good.

2

Subtract

Answer

Example 2: Coefficients again

Intermediate

Simplify:

1

Divide Numbers

Coefficients are normal numbers. Divide them.

2

Subtract Powers

Answer

Example 3: Variables Canceling Out

Advanced

Simplify:

1

Coefficients

2

Variable A

. The 's cancel out completely!

3

Variable B

Answer

Common Pitfalls

Dividing Exponents

Don't divide the powers! is , not .

Subtracting Coefficients

Don't subtract the big numbers. is , not .

Real-Life Applications

Sound Engineering:

  • The Decibel scale is logarithmic, which involves division of powers of 10.
  • Calculating signal loss often involves the quotient rule.

Practice Quiz

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