Absolutely! Let’s solve each prime factor decomposition problem using the factor tree method, as shown in the example. We’ll fill in the blanks in the trees and write the final answer in
index notation (also called exponential form).
---
🔢 Example (Given for reference):
16
- 16 = 2 × 8
- 8 = 2 × 4
- 4 = 2 × 2
→ All factors are 2 →
16 = 2⁴
---
##
✔ Problem 1: 36
Start with 36.
We can split 36 into:
- 36 =
2 × 18 (2 is prime, 18 is not)
- 18 =
2 × 9 (2 is prime, 9 is not)
- 9 =
3 × 3 (both are prime)
So the factor tree looks like:
```
36
/ \
2 18
/ \
2 9
/ \
3 3
```
All prime factors:
2, 2, 3, 3
In index notation:
36 = 2² × 3²
✔ Final Answer:
36 = 2² × 3²
---
##
✔ Problem 2: 54
Start with 54.
Split 54 into:
- 54 =
2 × 27 (2 is prime, 27 is not)
- 27 =
3 × 9 (3 is prime, 9 is not)
- 9 =
3 × 3 (both prime)
Factor tree:
```
54
/ \
2 27
/ \
3 9
/ \
3 3
```
Prime factors:
2, 3, 3, 3
In index notation:
54 = 2 × 3³
✔ Final Answer:
54 = 2 × 3³
---
##
✔ Problem 3: 135
Start with 135.
Split 135 into:
- 135 =
5 × 27 (5 is prime, 27 is not)
- 27 =
3 × 9 (3 is prime, 9 is not)
- 9 =
3 × 3
Factor tree:
```
135
/ \
5 27
/ \
3 9
/ \
3 3
```
Prime factors:
5, 3, 3, 3
In index notation:
135 = 3³ × 5
✔ Final Answer:
135 = 3³ × 5
---
##
✔ Problem 4: 525
Start with 525.
Split 525 into:
- 525 =
5 × 105 (5 is prime, 105 is not)
- 105 =
5 × 21 (5 is prime, 21 is not)
- 21 =
3 × 7 (both prime)
Factor tree:
```
525
/ \
5 105
/ \
5 21
/ \
3 7
```
Prime factors:
5, 5, 3, 7
In index notation:
525 = 3 × 5² × 7
✔ Final Answer:
525 = 3 × 5² × 7
---
##
✔ Problem 5: 693
Start with 693.
Split 693 into:
- 693 =
3 × 231 (3 is prime, 231 is not)
- 231 =
3 × 77 (3 is prime, 77 is not)
- 77 =
7 × 11 (both prime)
Factor tree:
```
693
/ \
3 231
/ \
3 77
/ \
7 11
```
Prime factors:
3, 3, 7, 11
In index notation:
693 = 3² × 7 × 11
✔ Final Answer:
693 = 3² × 7 × 11
---
## 📝 Summary of All Answers:
| Number | Prime Factor Decomposition (Index Notation) |
|--------|---------------------------------------------|
| 16 | 2⁴ |
| 36 | 2² × 3² |
| 54 | 2 × 3³ |
| 135 | 3³ × 5 |
| 525 | 3 × 5² × 7 |
| 693 | 3² × 7 × 11 |
---
## 💡 Tip for Factor Trees:
Always start by dividing by the smallest prime number (2, then 3, then 5, etc.) that divides the number evenly. Keep breaking down non-prime numbers until all branches end in prime numbers.
You’re now ready to fill in the worksheet with confidence! 🎯
Parent Tip: Review the logic above to help your child master the concept of prime factorization worksheet.
