006i - Exponent Notation for Prime Factorisation

Lesson 6 — Exponent Notation for Prime Factorisation

Strand: Number | Descriptor: AC9M7N02 | Duration: 45 minutes

Learning Intentions

• To represent natural numbers as products of powers of primes using exponent notation.
• To move fluently between expanded product form and index form.

Success Criteria

I can:

1. Rewrite a repeated prime product in exponent form, e.g. .
2. Evaluate a number given in index form.
3. Order the primes correctly and use the exponent to count repeats.
4. Use index form to compare the structure of two numbers.

Warmup

(5 minutes — rapid recall)

1. Evaluate: , , , , .
2. Write in exponent form: , , .
3. Which is larger, or ? Justify.

Answers: 1. ; 2. , , ; 3. . The base and exponent are not interchangeable.

Activities

Activity 1 — Explicit Instruction: Condensing to Index Form (12 min)

I do: Take last lesson’s result for .

Procedure to make explicit:

1. Prime factorise the number.
2. Sort the primes in ascending order.
3. Count how many times each prime appears — that count becomes the exponent.
4. Drop any exponent of (conventional, though not wrong to keep).

We do: Convert to index form.

You do: Express in index form: , , , , .

Activity 2 — Reversing: Index Form to Numeral (8 min)

Evaluate each, showing the expansion first:

Sequencing tip to model: evaluate each power first, then multiply — never multiply base by exponent.

Activity 3 — Inquiry: Reading Structure from Index Form (10 min)

Pairs. Calculators off — this is about reasoning, not arithmetic.

Two numbers are given only in index form:

Without fully evaluating either:

1. Which is larger? How can you tell?
2. Is divisible by ? By ? By ? Explain each.
3. How many factors does each have?

Socratic scaffolding:

Prompt	Purpose
What do and have in common?	Both contain .
So what is different?	has an extra ; has an extra .
Can you compare using only the difference?	and , so .
For divisibility — what does “divisible by ” require?	The factorisation must contain at least .
Does contain ?	Yes, has . So is divisible by .
What about ?	contains and , so yes.
Looking back — state a general rule.	divides exactly when every prime power in appears in with at least the same exponent.

Factor counts:

(Check: , . Both have 24 factors — a nice surprise worth discussing.)

Checks for Understanding

(5 minutes — exit ticket)

1. Write as a product of powers of primes.
2. Evaluate .
3. A student writes . Explain the error.
4. Is divisible by ? Justify without evaluating.
5. How many factors does have?

Answers: 1. ; 2. ; 3. Bases cannot be combined, and exponents are not added across different bases — , whereas ; 4. Yes — and the factorisation contains , which includes ; 5. .

Common Misconceptions

Misconception	How to pre-empt it
(combining unlike bases).	Evaluate both sides on the board. Rule: index laws only apply when the bases are the same.
(base times exponent).	Insist on the expansion line before evaluating, every time.
Miscounting repeats, e.g. for .	Have students tally each prime under the expansion before condensing.
Writing exponents in an arbitrary order, e.g. .	Convention: ascending prime order. Makes comparison between numbers immediate.
Believing .	Same base — the exponents add: . Verify numerically.
Assuming a larger exponent always means a larger number.	Compare with .

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Which is larger: or ?

Answer

and , so is larger — but only just. This near-equality is why computer scientists say a kilobyte is “about a thousand” bytes.

E2 (AMC Junior style). If , find and .

Answer

So , .

E3 (Challenge). What is the smallest number with exactly factors?

Answer

We need exponents . Options: ; ; ; ; gives only 9. Smallest is .

E4 (Challenge). The number . How many of ‘s factors are even?

Answer

Total factors: . Odd factors use no s: . So even factors: .

Homework

1. Write in index form: (a) (b) (c) .
2. Express as a product of powers of primes: (a) (b) (c) (d) (e) .
3. Evaluate: (a) (b) (c) (d) .
4. Find the number of factors of: (a) (b) (c) .
5. Reasoning. Without evaluating, explain how you know is divisible by .
6. Challenge. , where is a prime greater than . If , list all possible values of .

Answers: Q1 — (a) (b) (c) . Q2 — (a) (b) (c) (d) (e) . Q3 — (a) (b) (c) (d) . Q4 — (a) (b) (c) . Q5 — , and contains both and . Q6 — gives , so , i.e. or .