Integer Exponents: Zero, Negative, Combined

Lesson 2 of 2

In this lesson:

• Derive the zero and negative exponent rules from patterns
• Chain multiple exponent rules to simplify expressions

Learning Objectives for Lesson 2

By the end of this lesson, you should be able to:

3. Explain why for any nonzero base using the pattern of decreasing exponents
4. Explain why and convert expressions with negative exponents to positive-exponent form
5. Combine multiple exponent properties in sequence to simplify numerical expressions

Open Questions from Lesson 1

Two questions left unanswered:

• — so must equal ?
• — what does that mean?

Both answers come from one simple pattern.

Dividing by the Base Reveals the Pattern

Each row divides by 2 — the pattern forces , then , ,

Why Any Nonzero Base to Zero Equals One

Two arguments, same answer:

1. Pattern: — one more step past gives

2. Quotient rule: and — so

Both arguments require .

Check-In: Zero Exponent Boundary Cases

Evaluate each:

3. Is defined?

Think before you answer — especially the last one.

Pattern Continues Below Zero: Negative Exponents

The pattern continues past :

The rule:

A negative exponent means "take the reciprocal" — not "make it negative."

Negative Exponent Does Not Mean Negative Result

• Negative exponent: — the result is positive
• Negative base: — the result is negative

A negative exponent means reciprocal, not negative.

Converting Three Types of Negative Exponent Expressions

Write each with positive exponents:

Note the last one: — the fraction flips

Check-In: Apply the Negative Exponent Rule

Write with positive exponents, then evaluate:

For problem 2: apply the product rule first, then convert.

Consistency Check: Rules Still Work

Verify that :

By the product rule: ✓

By arithmetic: ✓

Both agree — our definitions of and keep all rules consistent.

All Seven Rules Ready to Chain Together

You now know all seven exponent properties:

1. Product of powers   5. Power of a quotient
2. Quotient of powers   6. Zero exponent
3. Power of a power    7. Negative exponent
4. Power of a product

The standard requires applying two or more rules in sequence.

Strategy: Identify Before You Compute

Before simplifying, ask:

1. What structure? — product, quotient, power of a power?
2. Which rule applies first? — name it, then execute
3. Repeat — ask again after each step

Simplify inside grouping symbols first (order of operations).

The CCSS Example: Two Rules in Sequence

Simplify :

Step 1 — Product of powers (same base, add exponents):

Step 2 — Negative exponent rule (convert to positive):

Two rules, two steps. Final answer:

Problem 1: Numerator First, Then Quotient

Simplify

Step 1 — Product of powers in numerator:

Step 2 — Quotient of powers:

Problem 2: Power of a Power, Then Quotient

Simplify

Step 1 — Power of a power (multiply exponents):

Step 2 — Quotient of powers (subtract exponents):

Problem 3: Two Solution Paths

Simplify

Path A:

Path B:

Both paths give ✓

Problem 4: Fraction with Negative Exponent

Simplify

Step 1 — Negative exponent on a fraction (flip):

Step 2 — Power of a quotient:

Step 3 — Zero exponent:

Final:

Practice: Chain Rules to Simplify Expressions

Simplify each expression. Show each rule you apply.

Identify which rule applies at each step before computing.

Practice: Rule Sequences and Final Simplified Answers

1. Option A: Inside first →

   Option B: Distribute → ✓

Summary: All Seven Exponent Rules

, not — negative exponent ≠ negative result

, not — zero exponent ≠ zero result

Product/quotient rules require same base —

(add), but (multiply)

What Comes Next: Scientific Notation

All seven rules appear in scientific notation operations:

• — power of a product + power of a power
• — product of powers

Next lesson: apply these rules to very large and very small numbers.