This worksheet guides students through prime factor decomposition using visual factor trees, complete with an example for the number 16.

Absolutely! Let's solve each prime factor decomposition problem using the factor tree method, as shown in the example. The goal is to break down each number into its prime factors (numbers divisible only by 1 and themselves) and then write the result in index notation (using exponents).

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## 🔢 Example: 16
As shown:
- 16 → 2 × 8
- 8 → 2 × 4
- 4 → 2 × 2
→ All factors are 2 → 16 = 2 × 2 × 2 × 2 = 2⁴

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# ✔ Now let’s solve the rest:

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## 🌳 1. 36

Start with 36.

- 36 → 2 × 18 *(2 is prime, 18 is not)*
- 18 → 2 × 9 *(2 is prime, 9 is not)*
- 9 → 3 × 3 *(both 3s are prime)*

So: 36 = 2 × 2 × 3 × 3
→ In index notation: 2² × 3²

✔ Answer: 36 = 2² × 3²

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## 🌳 2. 54

Start with 54.

- 54 → 2 × 27 *(2 is prime, 27 is not)*
- 27 → 3 × 9 *(3 is prime, 9 is not)*
- 9 → 3 × 3 *(both 3s are prime)*

So: 54 = 2 × 3 × 3 × 3
→ In index notation: 2 × 3³

✔ Answer: 54 = 2 × 3³

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## 🌳 3. 135

Start with 135.

- 135 → 5 × 27 *(5 is prime, 27 is not)*
- 27 → 3 × 9 *(3 is prime, 9 is not)*
- 9 → 3 × 3 *(both 3s are prime)*

So: 135 = 5 × 3 × 3 × 3
→ In index notation: 3³ × 5

✔ Answer: 135 = 3³ × 5

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## 🌳 4. 525

Start with 525.

- 525 → 5 × 105 *(5 is prime, 105 is not)*
- 105 → 5 × 21 *(5 is prime, 21 is not)*
- 21 → 3 × 7 *(both 3 and 7 are prime)*

So: 525 = 5 × 5 × 3 × 7
→ In index notation: 3 × 5² × 7

✔ Answer: 525 = 3 × 5² × 7

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## 🌳 5. 693

Start with 693.

Check divisibility:

- 693 ÷ 3 = 231 → so 693 = 3 × 231
- 231 ÷ 3 = 77 → so 231 = 3 × 77
- 77 = 7 × 11 → both prime

So: 693 = 3 × 3 × 7 × 11
→ In index notation: 3² × 7 × 11

✔ Answer: 693 = 3² × 7 × 11

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# 📝 Final Answers (for the worksheet boxes):

| Number | Prime Factor Decomposition (Index Notation) |
|--------|---------------------------------------------|
| 36 | 2² × 3² |
| 54 | 2 × 3³ |
| 135 | 3³ × 5 |
| 525 | 3 × 5² × 7 |
| 693 | 3² × 7 × 11 |

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## 💡 Tip for Factor Trees:
Always start with the smallest prime factor (usually 2, then 3, 5, 7, etc.) to make it easier. Keep breaking down non-prime numbers until all branches end in primes!

Let me know if you’d like to see the completed factor trees drawn out visually too! 😊