Free to do factor trees worksheet, Download Free to do factor ... - Free Printable

Absolutely! Let’s solve each factor tree step by step and write the prime factorization in both expanded form (product of primes) and exponential form.

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🔢 Example: 16

As shown:
- 16 = 2 × 8
- 8 = 2 × 4
- 4 = 2 × 2
→ All factors are prime: 2, 2, 2, 2
→ 16 = 2 × 2 × 2 × 2 = 2⁴

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## ✔ Problem 1: 36

Start with 36.

We can split 36 into:
- 36 = 2 × 18 → 2 is prime (blue circle), 18 is not (orange box)
- 18 = 2 × 9 → 2 is prime, 9 is not
- 9 = 3 × 3 → both 3s are prime

✔ So, prime factors: 2, 2, 3, 3
→ 36 = 2 × 2 × 3 × 3 = 2² × 3²

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## ✔ Problem 2: 54

Start with 54.

Split 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

✔ Prime factors: 2, 3, 3, 3
→ 54 = 2 × 3 × 3 × 3 = 2 × 3³

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## ✔ Problem 3: 135

Start with 135.

Split 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

✔ Prime factors: 5, 3, 3, 3
→ 135 = 3 × 3 × 3 × 5 = 3³ × 5

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## ✔ Problem 4: 525

Start with 525.

Split 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

✔ Prime factors: 5, 5, 3, 7
→ 525 = 3 × 5 × 5 × 7 = 3 × 5² × 7

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## ✔ Problem 5: 693

Start with 693.

Split 693:
- 693 = 3 × 231 → 3 is prime, 231 is not
- 231 = 3 × 77 → 3 is prime, 77 is not
- 77 = 7 × 11 → both 7 and 11 are prime

✔ Prime factors: 3, 3, 7, 11
→ 693 = 3 × 3 × 7 × 11 = 3² × 7 × 11

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## 📝 Final Answers (to fill in the worksheet):

| Number | Expanded Form | Exponential Form |
|--------|---------------------------|------------------------|
| 36 | 2 × 2 × 3 × 3 | 2² × 3² |
| 54 | 2 × 3 × 3 × 3 | 2 × 3³ |
| 135 | 3 × 3 × 3 × 5 | 3³ × 5 |
| 525 | 3 × 5 × 5 × 7 | 3 × 5² × 7 |
| 693 | 3 × 3 × 7 × 11 | 3² × 7 × 11 |

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## 💡 Tips for Factor Trees:

- Always start with the smallest prime factor (usually 2, then 3, 5, 7, etc.).
- Keep breaking down non-prime numbers until all branches end in prime numbers.
- Count repeated primes and write them as exponents for exponential form.

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