Prime factor decomposition worksheet using factor trees for numbers 16, 36, 54, 135, 525, and 693.

Let's solve each of the factor trees in Section A step by step, finding the prime factor decomposition of each number and expressing it in index notation.

We'll follow the same method as shown in the example:

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🔹 1. 36

Start with 36.

- 36 = 2 × 18 → 2 is prime ✔
- 18 = 2 × 9 → 2 is prime ✔
- 9 = 3 × 3 → both 3s are prime ✔

So:
- Prime factors: 2, 2, 3, 3
- In index form: $ 2^2 \times 3^2 $

✔ Fill in the tree:
```
36
/ \
2 18
/ \
2 9
/ \
3 3
```

Answer:
36 = 2² × 3²

---

🔹 2. 54

Start with 54.

- 54 = 2 × 27 → 2 is prime ✔
- 27 = 3 × 9 → 3 is prime ✔
- 9 = 3 × 3 → both 3s are prime ✔

So:
- Prime factors: 2, 3, 3, 3
- In index form: $ 2 \times 3^3 $

✔ Fill in the tree:
```
54
/ \
2 27
/ \
3 9
/ \
3 3
```

Answer:
54 = 2 × 3³

---

🔹 3. 135

Start with 135.

- 135 = 5 × 27 → 5 is prime ✔
- 27 = 3 × 9 → 3 is prime ✔
- 9 = 3 × 3 → both 3s are prime ✔

So:
- Prime factors: 5, 3, 3, 3
- In index form: $ 3^3 \times 5 $

✔ Fill in the tree:
```
135
/ \
5 27
/ \
3 9
/ \
3 3
```

Answer:
135 = 3³ × 5

---

🔹 4. 525

Start with 525.

- 525 = 5 × 105 → 5 is prime ✔
- 105 = 5 × 21 → 5 is prime ✔
- 21 = 3 × 7 → both 3 and 7 are prime ✔

So:
- Prime factors: 5, 5, 3, 7
- In index form: $ 3 \times 5^2 \times 7 $

✔ Fill in the tree:
```
525
/ \
5 105
/ \
5 21
/ \
3 7
```

Answer:
525 = 3 × 5² × 7

---

🔹 5. 693

Start with 693.

- 693 = 3 × 231 → 3 is prime ✔
- 231 = 3 × 77 → 3 is prime ✔
- 77 = 7 × 11 → both 7 and 11 are prime ✔

So:
- Prime factors: 3, 3, 7, 11
- In index form: $ 3^2 \times 7 \times 11 $

✔ Fill in the tree:
```
693
/ \
3 231
/ \
3 77
/ \
7 11
```

Answer:
693 = 3² × 7 × 11

---

✔ Final Answers (in index notation):

| Number | Prime Factor Decomposition |
|--------|----------------------------|
| 36 | $ 2^2 \times 3^2 $ |
| 54 | $ 2 \times 3^3 $ |
| 135 | $ 3^3 \times 5 $ |
| 525 | $ 3 \times 5^2 \times 7 $ |
| 693 | $ 3^2 \times 7 \times 11 $ |

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📝 How to fill in the factor trees:

For each tree, place:
- Blue circles: Prime numbers (e.g., 2, 3, 5, 7, etc.)
- Orange boxes: Composite numbers (non-prime) that need to be broken down further

> Example: For 36:
> - Top: 36 → split into 2 (blue) and 18 (orange)
> - 18 → split into 2 (blue) and 9 (orange)
> - 9 → split into 3 (blue) and 3 (blue)

You can now complete the worksheet accordingly!

Let me know if you'd like a printable version or visual diagram!