003i - The Zero Exponent

Lesson 3 — The Zero Exponent

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

Learning Intentions

• To understand what means and why for any nonzero base .
• To derive the zero-exponent law from a numerical pattern and from the division law.
• To apply the zero-exponent law within larger simplifications.

Success Criteria

I can:

1. State that any nonzero number raised to the power of equals .
2. Derive using the pattern of halving/thirding powers and using the division law.
3. Apply the zero-exponent law within expressions that combine multiple exponent laws.
4. Explain why must be nonzero for to make sense.

Warmup

(5 minutes — mini whiteboards, pattern spotting)

Complete the pattern:

1. What operation takes you from each value to the next, as the exponent decreases by ?
2. Apply that same operation to predict .
3. Now try the same pattern starting from . What do you predict for ?
4. Do you think this pattern will hold for any base? What about — can you halve your way down to it?

Teacher note: Question 4 is the hook. Do not resolve the case — flag it for the Common Misconceptions discussion; at this level is treated as undefined/excluded.

Activities

Activity 1 — Explicit Instruction: Deriving (10 min)

I do — Method 1: the pattern. Continue the warmup pattern explicitly: each step divides by the base.

I do — Method 2: the division law. Apply the division law from Lesson 1 to a power divided by itself.

Since both methods describe the same quantity, . State the general law:

We do: Together derive and using Method 2.

You do: Evaluate: , , , .

(Answers: all equal .)

Activity 2 — Guided Practice: Zero Exponent inside Mixed Expressions (10 min)

I do: Model a mixed expression where the zero exponent appears after simplifying.

We do: Together simplify and .

You do: Simplify:

(Answers: 1. ; 2. ; 3. — any nonzero base raised to a power that simplifies to exponent gives , regardless of how “big” the base looks.)

Activity 3 — Applied Task: the Zero Exponent in Context (14 min)

Pairs.

Problem 1. A researcher records bacteria population as , where is the number of hours elapsed. What is the population at ? Use the zero-exponent law to explain what this tells you about the population at the very start.

Problem 2. Simplify fully. What does the result tell you about the relationship between and ?

Problem 3 (harder). Without evaluating fully, decide whether equals , and explain what that means about the numerator and denominator.

Socratic scaffolding for Problem 3:

Prompt	Purpose
Understand: what is the expression asking?	Whether the top and bottom of the fraction are, in fact, equal.
Devise a plan	Simplify the denominator using the multiplication law first, then apply the division law.
Carry out the plan	, so the fraction becomes .
What does an exponent of mean here?	The numerator and denominator are exactly equal, since anything divided by itself gives .
Looking back	Check directly: , and too. The zero exponent correctly signals “these are the same value.”

Answers: Problem 1 — ; this tells us the starting (initial) population is , before any growth has occurred. Problem 2 — ; this tells us and are exactly equal in value. Problem 3 — Yes, it equals , meaning the numerator and denominator are equal.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Evaluate .
2. Simplify .
3. Explain, using the division law, why for any nonzero .
4. Is equal to ? Explain why this case is treated differently.

Answers: 1. ; 2. ; 3. , and any nonzero number divided by itself equals , so must equal ; 4. is not evaluated using this reasoning, because it would require dividing by , i.e. dividing by , which is undefined. is excluded/treated as undefined at this level.

Common Misconceptions

Misconception	How to pre-empt it
Believing .	Directly contrast: "" gives , but "" gives — these are different operations entirely. Reinforce with the division-law derivation each time.
Believing a “bigger” base or exponent inside brackets should give a bigger answer even when the overall exponent is , e.g. assuming .	Emphasise that the base itself never matters once the exponent is — only whether the base is nonzero.
Applying even when .	Explicitly flag as an excluded/undefined case at this level; do not let students generalise “any base” without the nonzero condition.
Confusing with , e.g. thinking .	Drill the distinction: (one factor of ), (no factors of — the “empty product”).
Forgetting to simplify to first within a longer expression, and instead cancelling terms incorrectly.	Insist that every division-to-equal-exponents step is written out explicitly as before evaluating.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Evaluate .

Answer

Each term equals (since each base is nonzero), so the sum is .

E2 (Kangaroo style). If , and is a positive integer, find .

Answer

For to equal , the exponent must be (since ):

E3 (Challenge). Simplify for any nonzero , and explain your result.

Answer

The numerator and denominator simplify to the same power, so the expression always equals , regardless of the value of (as long as ).

Homework

1. Evaluate: (a) (b) (c) (d) .
2. Simplify: (a) (b) (c) .
3. A student claims “because anything to the power of zero is nothing.” Explain why this reasoning is incorrect.
4. Reasoning. Use the division law to explain why works for but the same reasoning breaks down for .
5. Challenge. Find all positive integer values of for which .

Answers: 1(a) (b) (c) (d) . 2(a) (b) (c) . 3. Incorrect — is not “nothing,” it equals ; this follows from , and any nonzero number divided by itself is , not . 4. For : , matching . For : would require dividing by , which is undefined, so the reasoning cannot be applied and is excluded. 5. is the only solution, since requires .