001i - Multiplying and Dividing Powers

Lesson 1 — Multiplying and Dividing Powers with the Same Base

Strand: Number | Descriptor: AC9M8N02 | Duration: 45 minutes

Learning Intentions

• To understand exponent notation and what the base and exponent represent.
• To establish and apply the multiplication law for powers with the same base.
• To establish and apply the division law for powers with the same base.

Success Criteria

I can:

1. Explain what the base and exponent represent in .
2. Apply to simplify a product of powers with the same base.
3. Apply to simplify a quotient of powers with the same base.
4. Justify each law by expanding powers into repeated factors.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

1. Write as repeated multiplication, then evaluate it.
2. In , name the base and the exponent.
3. Evaluate and separately, then multiply the two results. What single power of gives the same answer?
4. True or false: can be simplified to a single power. Explain your answer.

Teacher note: Question 3 previews the multiplication law developed in Activity 1. Question 4 previews the “same base” restriction — do not resolve it yet, but flag both answers for discussion.

Activities

Activity 1 — Explicit Instruction: Multiplying Powers with the Same Base (10 min)

I do: Expand fully as repeated factors and count them.

Say aloud: “Three factors of , then four more factors of — that’s seven factors of altogether. The base stays the same; the exponents add.” State the general law:

We do: Together simplify and , expanding first, then applying the law directly.

You do: Simplify, leaving each answer as a single power: , , .

(Answers: ; — remind students that means ; .)

Activity 2 — Explicit Instruction: Dividing Powers with the Same Base (10 min)

I do: Expand as a fraction and cancel common factors.

State the general law:

We do: Together simplify and .

You do: Simplify: , , .

(Answers: ; ; — do not evaluate this last one yet. Flag it: “What might a power with an exponent of zero be worth? We’ll answer that properly in Lesson 3.“)

Activity 3 — Applied Task: Combining both Laws (14 min)

Pairs.

Problem 1. A server’s total storage capacity is bytes. A single photo file uses bytes of that storage.

(a) Simplify to find, as a single power of , how the total capacity compares to one photo’s size.

(b) Four identical servers, each with capacity bytes, are combined so their capacities multiply the number of possible file arrangements by per extra server connected. Simplify .

Problem 2 (harder — compare without a calculator). Which is greater: , or ?

Socratic scaffolding for Problem 2:

Prompt	Purpose
Understand: what is being compared?	Two products of powers of — not their full evaluated values.
Devise a plan: what tool do you have?	The multiplication law lets you rewrite each product as a single power of .
Carry out the plan for the first expression	.
Carry out the plan for the second expression	.
Now compare	Same base, so the larger exponent gives the larger value: .
Looking back — does this make sense?	, so the second expression is exactly double the first. Check with a rough estimate: doubling something is a believable difference for one extra factor of in the exponent.

Since , the second expression, , is greater.

Answers to Problem 1: (a) ; (b) .

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Simplify , leaving your answer as a single power.
2. Simplify , leaving your answer as a single power.
3. Explain why cannot be combined into a single power using the multiplication law.
4. Without fully evaluating either expression, state which is greater: or . Justify your answer.

Answers: 1. ; 2. ; 3. The bases are different ( and ) — the law only applies when the base is identical; 4. and , so the second is greater since .

Common Misconceptions

Misconception	How to pre-empt it
Adding the bases as well as the exponents: .	Insist on the sentence stem “keep the base, add the exponents” every time the law is used.
Applying the law across different bases: .	Contrast directly with a same-base example side by side; require students to check “same base?” before applying any law.
Multiplying the exponents instead of adding them: .	Verify numerically: , not . This confusion previews and is resolved fully in Lesson 2 (power of a power).
Reversing the order in the division law: .	Anchor to the expanded-fraction method — the dividend’s exponent always comes first in the subtraction.
Forgetting the invisible exponent of : treating as .	Explicitly rewrite as before applying any law, every time a “bare” base appears.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Simplify and evaluate .

Answer

E2 (Kangaroo style). If , find .

Answer

So .

E3 (Challenge). Which is larger, or ?

Answer

They are equal. This uses a different index law — powers of a product: . So . Note this is not the same-base law from today’s lesson, since and have different bases — it is a useful extension to notice.

Homework

1. Simplify, leaving each answer as a single power: (a) (b) (c) (d) .
2. Evaluate: (a) (b) .
3. A student wrote . Is this correct? Explain the error, if any, and give the correct answer.
4. Reasoning. Using expanded (repeated-factor) form, explain why is true for any positive integers and .
5. Challenge. Find the value of such that .

Answers: 1(a) (b) (c) (d) . 2(a) (b) . 3. Incorrect — exponents are added, not multiplied, when the base stays the same: , not . 4. Expanding gives factors of , and gives factors of ; multiplying them together gives factors of in total, which is . 5. .