Prime Factorization Trees Answer Key showing factorization of numbers 4, 14, and 75.

You're looking at a Prime Factorization Trees worksheet designed to help students break down composite numbers into their prime factors using factor trees, and then express the result in exponential form.

Let’s solve each problem step by step and explain the reasoning.

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🔢 Problem 1: Number = 4

Factor Tree:

```
4
/ \
2 2
```

- 4 can be broken into 2 × 2.
- Both 2s are prime numbers (they can’t be factored further).
- So, prime factorization of 4 is 2 × 2.
- In exponential form: 2²

✔ Answer: 2²

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🔢 Problem 2: Number = 14

Factor Tree:

```
14
/ \
2 7
```

- 14 = 2 × 7
- 2 and 7 are both prime numbers.
- No repeated factors → no exponent needed beyond 1.
- Prime factorization: 2 × 7
- In exponential form: 2¹ × 7¹, but typically we omit the exponent “1”.

✔ Answer: 2 × 7

*(Note: The worksheet shows “2×7” — correct. Some might write it as 2¹×7¹, but standard practice is to omit exponent 1.)*

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🔢 Problem 3: Number = 75

Factor Tree:

```
75
/ \
3 25
/ \
5 5
```

- Start with 75 = 3 × 25
- Then 25 = 5 × 5
- All factors now: 3, 5, 5 → all prime!
- So prime factorization: 3 × 5 × 5
- Group repeated factors: 5 appears twice → 5²
- Exponential form: 3 × 5²

✔ Answer: 3 × 5²

*(The worksheet already shows this correctly.)*

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## ✔ Final Answers (as requested):

| Number | Prime Factorization (Exponential Form) |
|--------|----------------------------------------|
| 4 | 2² |
| 14 | 2 × 7 |
| 75 | 3 × 5² |

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## 🧠 Explanation Summary:

- Prime Factorization means breaking a number down into a product of prime numbers only.
- A factor tree helps visualize the breakdown.
- Exponential form is used when a prime factor repeats — you write the base and raise it to the power of how many times it appears.
- Example: 2 × 2 × 2 = 2³
- If a prime appears only once, you don’t need to write the exponent “1” — just write the prime.

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✔ You’ve now successfully completed the worksheet! Let me know if you want to try more examples or learn about GCF/LCM using prime factorization.