Properties of Integer Exponents

Lesson 1 of 2

In this lesson:

• Apply the product and quotient of powers rules
• Apply the power of a power, product, and quotient rules

Learning Objectives for Lesson 1

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

1. State and apply the product of powers rule —
2. State and apply the quotient of powers rule —
3. Apply the power of a power rule —
4. Apply the power of a product rule —
5. Apply the power of a quotient rule —

Do You Want to Write 27 Factors?

• means 15 factors of 2; means 12 more
• Writing out 27 twos is tedious — there must be a pattern

How many total factors of 2 do we have?

Expanding to Count All the Factors

How does 5 relate to 3 and 2? →

Product of Powers: The Rule

Two groups of factors merge into one — so we add the exponents.

Product of Powers: Quick Practice

Apply the rule to these:

Compute the exponent for each before the next slide.

Quick Check: Product of Powers

Simplify — which rule applies?

• Rule: product of powers
• ✓

Now try: what is ?

Hold that thought — we'll learn what means in Lesson 2.

Product and Quotient Rules Require Same Base

Can we apply the product rule to ?

• Different bases — factors of 2 and 5 cannot merge

Rule: requires the same base .

Quotient of Powers: Cancellation View

Start with :

Cancel two factors of 6 from numerator and denominator:

Quotient of Powers: The Rule

Practice:

Preview question: What is ?

By the rule: — what does a negative exponent mean?

Answer coming in Lesson 2.

Bridge to the Next Rules

We've covered:

• → add exponents
• → subtract exponents

New question: What about — a power raised to another power?

And what about — a product raised to a power?

Different structure → different rule.

Power of a Power: Expanding First

Apply the definition of the outer exponent — four copies of :

Apply the product of powers rule:

Rule: — multiply the exponents.

Critical Contrast: Add or Multiply?

• Two separate powers multiplied: → add
• One power raised to another: → multiply

Different structure → different operation.

Quick Check: Add or Multiply Exponents?

Name the rule, then simplify:

Look at the structure before computing.

Power of a Product: Expand and Verify

Verify: and ✓

Rule: — exponent distributes to each factor

Power of a Quotient Rule

Expand as repeated multiplication of fractions:

Rule:

The exponent distributes to numerator and denominator.

Combining Power Rules: Worked Example

Simplify

Path A:

• and

Path B:

• Distribute outer exponents first:

Both paths give ✓

Practice: Add, Multiply, or Distribute?

For each expression, name the rule and simplify:

Identify the structure before you compute.

Practice: Rule Names and Simplified Answers

1. — power of a power (multiply)
2. — power of a product
3. — power of a quotient

Summary: Five Exponent Rules Covered

(add) but (multiply)

Same base required for product and quotient rules

Next Up: Zero and Negative Exponents

Two open questions from today:

1. — so must equal
2. — what does that mean?

Lesson 2: derive both rules from patterns, then chain all seven rules.