002i - Power of a Power

Lesson 2 — The Power of a Power Law

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

Learning Intentions

• To understand what it means to raise a power to a further power.
• To establish and apply the power-of-a-power law .
• To combine the power-of-a-power law with the multiplication and division laws from Lesson 1.

Success Criteria

I can:

1. Explain what represents as repeated multiplication of .
2. Apply to simplify a power raised to a power.
3. Combine the power-of-a-power law with the multiplication and division laws in a single expression.
4. Justify the law by expanding it into repeated factors.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

1. Simplify (recap of Lesson 1).
2. Simplify (recap of Lesson 1).
3. Expand by writing out three times, multiplied together. Evaluate it.
4. Evaluate . What do you notice, compared to Question 3?

Teacher note: Question 4 is the hook — students should notice , since . Do not name the rule yet; let Activity 1 formalise it.

Activities

Activity 1 — Explicit Instruction: the Power-of-a-power Law (10 min)

I do: Expand as repeated multiplication of , then apply the multiplication law from Lesson 1 to finish.

Say aloud: “Four groups of two factors of — that’s eight factors of in total. Multiplying the exponents gives the same result as repeated addition.” State the general law:

Critical contrast to display side by side:

We do: Together simplify and .

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

(Answers: ; ; .)

Activity 2 — Guided Practice: Combining All Three Laws (10 min)

I do: Model a mixed expression, applying power-of-a-power first, then multiplication.

We do: Together simplify and .

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

(Answers: 1. ; 2. ; 3. .)

Activity 3 — Applied Task: Scaling Problems (14 min)

Pairs.

Problem 1. A cube-shaped storage block has a side length that is itself a power of : the side is units long.

(a) The volume of a cube is (side length). Write the volume as a power of , using the power-of-a-power law.

(b) A second block has side length units. Write its volume as a power of .

(c) How many of the smaller blocks would exactly fill the larger block? (Hint: divide the volumes.)

Problem 2 (harder — order matters). Is equal to ?

Socratic scaffolding for Problem 2:

Prompt	Purpose
Understand: what is different about the two expressions?	One is “a power, then raised to a power”; the other is “a base raised to a power-of-a-power exponent” — the brackets sit in different places.
Devise a plan	Evaluate each expression separately using the correct order of operations.
Carry out the plan for	, then . Or using the law: .
Carry out the plan for	first (innermost bracket), then .
Compare	— the two expressions are not equal.
Looking back	Why does the difference matter? multiplies the exponents (); has a “tower” of exponents that must be evaluated top-down, exponent first. Order of operations for towers works from the top exponent downward, not left to right.

Answers to Problem 1: (a) ; (b) ; (c) small blocks.

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 the difference between and , and evaluate both.
4. A student simplified as . Identify and correct the error.

Answers: 1. ; 2. ; 3. (multiply the exponents); (evaluate the top exponent first) — they are different; 4. The student added the exponents instead of multiplying them; the correct answer is .

Common Misconceptions

Misconception	How to pre-empt it
Adding the exponents instead of multiplying: .	Contrast directly with the multiplication law from Lesson 1 — display both laws side by side and ask “which operation joins the two powers here?”
Treating and as the same expression.	Use the “tower” language: unbracketed towers evaluate top-down; bracketed power-of-a-power multiplies exponents. Always test with small numbers when in doubt.
Applying the law when the base changes between the inner and outer power, e.g. read as .	Re-expand from first principles whenever students appear to guess rather than apply the law directly.
Forgetting to apply the power-of-a-power law before combining with multiplication/division laws in a mixed expression.	Model a consistent order: simplify any bracketed power-of-a-power terms first, then apply multiplication/division laws.
Believing .	Expand fully as three separate factors of to show it is repeated multiplication of the same power, not a single multiplication of two powers.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Simplify and evaluate.

Answer:

E2 (Kangaroo style). Find such that .

E3 (Challenge). Which is greater: or ?

They are equal: and . Multiplication of the exponents is commutative (), so swapping which exponent is “inner” and “outer” never changes the result.

E4 (Extension). Show that for any base , using the power-of-a-power law.

Both equal , confirming the two are always equal for the same reason as E3.

Homework

1. Simplify, leaving each answer as a single power: (a) (b) (c) (d) .
2. Evaluate: (a) (b) .
3. A student simplified as . Identify the error and give the correct simplification.
4. Reasoning. Explain, using repeated multiplication, why for positive integers and .
5. Challenge. Find the value of such that .

Answers: 1(a) (b) (c) (d) . 2(a) (b) . 3. The student added the exponents instead of multiplying them; correctly, . 4. means groups of , multiplied together; each group contributes factors of , giving factors of in total, which is . 5. .