Lesson 8.4

Zero & Negative Exponents

What does it mean to have negative amounts of multiplication? It means division.

Introduction

We have seen positive exponents (). But what about ? Or ? These aren't "negative numbers". They are instructions to Reciprocate.

Past Knowledge

Lesson 8.2 (Quotient Rule). Remember that .

Today's Goal

Recognize that and .

Future Success

Scientific Notation uses negative exponents to represent very small numbers (like the size of an atom).

Key Concepts

The Zero Exponent

Anything (except 0) raised to the zero power is ONE.

Why?

and

Negative Exponents

A negative exponent means "Wrong Side of the Fraction Bar".

If it's upstairs...

...move it downstairs.

If it's downstairs...

...move it upstairs.

Worked Examples

Example 1: The Zero Rule

Basic

Simplify:

1

Identify the Base

Only the has the zero power. The 5 is separate.

2

Simplify

Answer

Example 2: Moving Negatives

Intermediate

Simplify:

1

Find the Negatives

The is unhappy upstairs.

The and are fine.

2

Move It

Move to the bottom. The negative sign disappears upon moving.

Answer

Example 3: Coefficients Simplify

Advanced

Simplify:

1

Flip First

A negative exponent on the outside flips the ENTIRE fraction.

2

Fix Internal Negatives

That moves down.

3

Distribute Power

Square everything.

Answer

Common Pitfalls

Negative Number vs. Exponent

makes a fraction. makes a negative number. Don't confuse sign with position.

The Trapped Coefficient

In , the 5 does NOT move. Only the has the negative exponent.

Real-Life Applications

Science:

  • The mass of a Proton is kg.
  • Without negative exponents, we would have to write 0.000... (26 zeros) ...167. They keep numbers meaningful.

Practice Quiz

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