Australian Curriculum v9 / ACiQ Year 7 Mathematics - Unit 0 - Factors, prime factors, powers and square roots

Factors, prime factors, powers and square roots

Represent natural numbers with prime factors, exponents and powers of 10, and solve problems with perfect squares and square roots.

Updated 2026-07-24 - 7 min read

Represent natural numbers with prime factors, exponents and powers of 10, and solve problems with perfect squares and square roots. This draft teaching resource is aligned to the cited Year 7 content descriptions and still requires named human curriculum approval.

Track the value represented at every step: an efficient method is valid only when it preserves the number, comparison or required accuracy.

1. Factors, primes and prime factorisation

A prime has exactly two positive factors. Prime factorisation continues until every branch ends in a prime.

Decision check: Apply this lesson's rule to the worked example: Different factor-tree routes must end in the same prime factors. Identify the exact value, label or property that satisfies the rule.

2. Powers and expanded notation

In 3⁴, 3 is the repeated factor and 4 counts how many factors. In expanded notation, each digit is multiplied by its place-value power of 10.

Decision check: Apply this lesson's rule to the worked example: An exponent counts factors; it is not a multiplier. Identify the exact value, label or property that satisfies the rule.

3. Perfect squares and square roots

A perfect square is n × n for a natural number n. The positive square root reverses that operation.

Decision check: Apply this lesson's rule to the worked example: Check a square root by multiplying the proposed root by itself. Identify the exact value, label or property that satisfies the rule.

Worked examples with complete reasoning

Worked example 1

Question: Write 36 as a product of prime factors using exponent notation.

  1. Divide 36 by prime numbers until every factor is prime.
  2. Collect repeated factors: 2^2 × 3^2.

Answer: 2^2 × 3^2

Check: Start with the smallest prime divisor and continue until the quotient is 1.

Worked example 2

Question: Write 4507 in expanded notation using powers of 10. Omit zero place-value terms.

  1. Match each non-zero digit to its place value.
  2. The expanded form is 4 × 10^3 + 5 × 10^2 + 7.

Answer: 4 × 10^3 + 5 × 10^2 + 7

Check: The exponent on 10 matches the number of places to the right of the digit.

Worked example 3

Question: Find the positive square root of 64.

  1. Ask which positive number multiplied by itself equals 64.
  2. 8 × 8 = 64, so √64 = 8.

Answer: 8

Check: Recall nearby perfect squares.

Method map for this topic

  1. Factors, primes and prime factorisation: Different factor-tree routes must end in the same prime factors.
  2. Powers and expanded notation: An exponent counts factors; it is not a multiplier.
  3. Perfect squares and square roots: Check a square root by multiplying the proposed root by itself.

Common mistakes and how to correct them

  • prime factorisation: Not yet. Start with the smallest prime divisor and continue until the quotient is 1. Recheck the prime factorisation step, then try again.
  • exponent meaning: Not yet. An exponent counts repeated factors, not repeated addition. Recheck the exponent meaning step, then try again.
  • square root: Not yet. Recall nearby perfect squares. Recheck the square root step, then try again.
  • expanded notation: Not yet. The exponent on 10 matches the number of places to the right of the digit. Recheck the expanded notation step, then try again.
  • common factor: Not yet. Keep only prime factors shared by both numbers. Recheck the common factor step, then try again.
  • square context: Not yet. The side length of a square array is the square root of the total. Recheck the square context step, then try again.

Practice

  1. Write 36 as a product of prime factors using exponent notation.
  2. Write 72 as a product of prime factors using exponent notation.
  3. Write 4507 in expanded notation using powers of 10. Omit zero place-value terms.
  4. Write 60804 in expanded notation using powers of 10. Omit zero place-value terms.
  5. Find the positive square root of 64.
  6. Find the positive square root of 121.

<details> <summary>Answers and reasoning</summary>

  1. 2^2 × 3^2 — Divide 36 by prime numbers until every factor is prime. Collect repeated factors: 2^2 × 3^2.
  2. 2^3 × 3^2 — Divide 72 by prime numbers until every factor is prime. Collect repeated factors: 2^3 × 3^2.
  3. 4 × 10^3 + 5 × 10^2 + 7 — Match each non-zero digit to its place value. The expanded form is 4 × 10^3 + 5 × 10^2 + 7.
  4. 6 × 10^4 + 8 × 10^2 + 4 — Match each non-zero digit to its place value. The expanded form is 6 × 10^4 + 8 × 10^2 + 4.
  5. 8 — Ask which positive number multiplied by itself equals 64. 8 × 8 = 64, so √64 = 8.
  6. 11 — Ask which positive number multiplied by itself equals 121. 11 × 11 = 121, so √121 = 11.

</details>

Checklist and vocabulary

| Skill | Observable evidence | | --- | --- | | represent prime factorisation exponents | Write natural numbers as products of prime powers using exponent notation. | | use expanded notation powers ten | Represent natural numbers in expanded notation with powers of 10. | | solve perfect square root problems | Connect perfect squares with positive square roots in problems. |

Retrieval prompts

  • State the defining relationship for each method above without looking.
  • Give one example where two visually similar representations mean different things.
  • Diagnose one misconception from the list and explain the first corrective step.
  • Name a check that is independent of simply repeating the same calculation.

Teach-back and self-check

Close the worked examples and explain factors, prime factors, powers and square roots as if you were helping another Year 7 student. Begin by naming the clue that tells you which relationship, representation or operation is needed. Then show one complete example with labels and units where they matter. At each line, explain what stayed equivalent and why the next step is allowed.

Use this four-part check before calling the method secure:

  1. Identify: What information is given, and what must be found or justified?
  2. Represent: Which number line, diagram, table, expression, graph or organised list makes the relationship visible?
  3. Reason: Which rule connects the representation to the required answer?
  4. Verify: Can you check by estimation, substitution, an inverse operation, a second representation or the original context?

If the explanation depends on “I just knew”, return to the method map and name the missing decision. A durable method is one you can explain, check and use in a changed context after a delay.

Transfer task

Create a factors, prime factors, powers and square roots decision from a shopping, recipe, travel or measurement context. State the quantities and required accuracy, solve it, then explain why your representation and rounding are suitable.

Sources