Algebraic expressions for simplification practice.
A list of 20 algebraic expressions involving variables a and b, presented in two columns, with equations numbered 1 to 20.
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Step-by-step solution for: Collecting like terms (with powers) - Variation Theory
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Show Answer Key & Explanations
Step-by-step solution for: Collecting like terms (with powers) - Variation Theory
You’ve uploaded an image containing 20 algebraic expressions, each requiring simplification by combining like terms.
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Like terms are terms that have the same variable(s) raised to the same power(s). For example:
- `a²` and `6a²` are like terms.
- `ab` and `3ab` are like terms.
- `a²b` and `5a²b` are like terms.
- But `a²` and `ab` are not like terms.
We combine them by adding or subtracting their coefficients.
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Let’s solve each one step-by-step:
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Group like terms:
→ `(a² + a²) + (b² + b²)` = `2a² + 2b²`
✔ Answer: `2a² + 2b²`
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→ `(6a² + a²) + (b² + b²)` = `7a² + 2b²`
✔ Answer: `7a² + 2b²`
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→ `(6a² + a²) + (5b² + b²)` = `7a² + 6b²`
✔ Answer: `7a² + 6b²`
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→ `(6a² + 4a²) + (5b² + b²)` = `10a² + 6b²`
✔ Answer: `10a² + 6b²`
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→ `(6a² + 4a²) + (5b² + 3b²)` = `10a² + 8b²`
✔ Answer: `10a² + 8b²`
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→ `(6a² - 4a²) + (5b² + 3b²)` = `2a² + 8b²`
✔ Answer: `2a² + 8b²`
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→ `(6a² - 4a²) + (5b² - 3b²)` = `2a² + 2b²`
✔ Answer: `2a² + 2b²`
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→ `(6a² - 4a²) + (-5b² - 3b²)` = `2a² - 8b²`
✔ Answer: `2a² - 8b²`
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→ `(6a² - 4a²) + (-5b - 3b)` = `2a² - 8b`
✔ Answer: `2a² - 8b`
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→ `(6a² - 4a²) + (-5ab - 3ab)` = `2a² - 8ab`
✔ Answer: `2a² - 8ab`
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Note: `ba = ab` (commutative property)
→ `(6a² - 4a²) + (-5ab - 3ab)` = `2a² - 8ab`
✔ Answer: `2a² - 8ab`
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Group like terms:
→ `(6a² - 4a²) + (-5a²b - 3a²b)` = `2a² - 8a²b`
✔ Answer: `2a² - 8a²b`
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→ `(6a² - 4a²) + (-5a²b + 3a²b)` = `2a² - 2a²b`
✔ Answer: `2a² - 2a²b`
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→ `(6a² - 4a²) + (-5a²b) + (3ab²)` = `2a² - 5a²b + 3ab²`
Note: `a²b` and `ab²` are not like terms — different powers!
✔ Answer: `2a² - 5a²b + 3ab²`
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→ `(6a² - 4a²) + (-5ab² + 3ab²)` = `2a² - 2ab²`
✔ Answer: `2a² - 2ab²`
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→ `(-6a² - 4a²) + (-5ab² + 3ab²)` = `-10a² - 2ab²`
✔ Answer: `-10a² - 2ab²`
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Constants: `-6 - 4 = -10`
`ab²` terms: `-5ab² + 3ab² = -2ab²`
✔ Answer: `-10 - 2ab²`
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→ `(-6a - 4a) + (-5ab² + 3ab²)` = `-10a - 2ab²`
✔ Answer: `-10a - 2ab²`
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→ `(-6ab - 4ab) + (-5ab² + 3ab²)` = `-10ab - 2ab²`
✔ Answer: `-10ab - 2ab²`
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→ `(-6a²b - 4a²b) + (-5ab² + 3ab²)` = `-10a²b - 2ab²`
✔ Answer: `-10a²b - 2ab²`
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## 📝 Final Answers Summary:
1. `2a² + 2b²`
2. `7a² + 2b²`
3. `7a² + 6b²`
4. `10a² + 6b²`
5. `10a² + 8b²`
6. `2a² + 8b²`
7. `2a² + 2b²`
8. `2a² - 8b²`
9. `2a² - 8b`
10. `2a² - 8ab`
11. `2a² - 8ab`
12. `2a² - 8a²b`
13. `2a² - 2a²b`
14. `2a² - 5a²b + 3ab²`
15. `2a² - 2ab²`
16. `-10a² - 2ab²`
17. `-10 - 2ab²`
18. `-10a - 2ab²`
19. `-10ab - 2ab²`
20. `-10a²b - 2ab²`
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✔ All problems solved by identifying and combining like terms. Remember: only terms with identical variable parts can be combined!
Let me know if you want these explained in video format or need worksheets! 😊
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✔ What is “combining like terms”?
Like terms are terms that have the same variable(s) raised to the same power(s). For example:
- `a²` and `6a²` are like terms.
- `ab` and `3ab` are like terms.
- `a²b` and `5a²b` are like terms.
- But `a²` and `ab` are not like terms.
We combine them by adding or subtracting their coefficients.
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Let’s solve each one step-by-step:
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🔢 1. `a² + b² + a² + b²`
Group like terms:
→ `(a² + a²) + (b² + b²)` = `2a² + 2b²`
✔ Answer: `2a² + 2b²`
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🔢 2. `6a² + b² + a² + b²`
→ `(6a² + a²) + (b² + b²)` = `7a² + 2b²`
✔ Answer: `7a² + 2b²`
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🔢 3. `6a² + 5b² + a² + b²`
→ `(6a² + a²) + (5b² + b²)` = `7a² + 6b²`
✔ Answer: `7a² + 6b²`
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🔢 4. `6a² + 5b² + 4a² + b²`
→ `(6a² + 4a²) + (5b² + b²)` = `10a² + 6b²`
✔ Answer: `10a² + 6b²`
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🔢 5. `6a² + 5b² + 4a² + 3b²`
→ `(6a² + 4a²) + (5b² + 3b²)` = `10a² + 8b²`
✔ Answer: `10a² + 8b²`
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🔢 6. `6a² + 5b² - 4a² + 3b²`
→ `(6a² - 4a²) + (5b² + 3b²)` = `2a² + 8b²`
✔ Answer: `2a² + 8b²`
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🔢 7. `6a² + 5b² - 4a² - 3b²`
→ `(6a² - 4a²) + (5b² - 3b²)` = `2a² + 2b²`
✔ Answer: `2a² + 2b²`
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🔢 8. `6a² - 5b² - 4a² - 3b²`
→ `(6a² - 4a²) + (-5b² - 3b²)` = `2a² - 8b²`
✔ Answer: `2a² - 8b²`
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🔢 9. `6a² - 5b - 4a² - 3b`
→ `(6a² - 4a²) + (-5b - 3b)` = `2a² - 8b`
✔ Answer: `2a² - 8b`
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🔢 10. `6a² - 5ab - 4a² - 3ab`
→ `(6a² - 4a²) + (-5ab - 3ab)` = `2a² - 8ab`
✔ Answer: `2a² - 8ab`
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🔢 11. `6a² - 5ab - 4a² - 3ba`
Note: `ba = ab` (commutative property)
→ `(6a² - 4a²) + (-5ab - 3ab)` = `2a² - 8ab`
✔ Answer: `2a² - 8ab`
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🔢 12. `6a² - 5a²b - 4a² - 3a²b`
Group like terms:
→ `(6a² - 4a²) + (-5a²b - 3a²b)` = `2a² - 8a²b`
✔ Answer: `2a² - 8a²b`
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🔢 13. `6a² - 5a²b - 4a² + 3a²b`
→ `(6a² - 4a²) + (-5a²b + 3a²b)` = `2a² - 2a²b`
✔ Answer: `2a² - 2a²b`
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🔢 14. `6a² - 5a²b - 4a² + 3ab²`
→ `(6a² - 4a²) + (-5a²b) + (3ab²)` = `2a² - 5a²b + 3ab²`
Note: `a²b` and `ab²` are not like terms — different powers!
✔ Answer: `2a² - 5a²b + 3ab²`
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🔢 15. `6a² - 5ab² - 4a² + 3ab²`
→ `(6a² - 4a²) + (-5ab² + 3ab²)` = `2a² - 2ab²`
✔ Answer: `2a² - 2ab²`
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🔢 16. `-6a² - 5ab² - 4a² + 3ab²`
→ `(-6a² - 4a²) + (-5ab² + 3ab²)` = `-10a² - 2ab²`
✔ Answer: `-10a² - 2ab²`
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🔢 17. `-6 - 5ab² - 4 + 3ab²`
Constants: `-6 - 4 = -10`
`ab²` terms: `-5ab² + 3ab² = -2ab²`
✔ Answer: `-10 - 2ab²`
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🔢 18. `-6a - 5ab² - 4a + 3ab²`
→ `(-6a - 4a) + (-5ab² + 3ab²)` = `-10a - 2ab²`
✔ Answer: `-10a - 2ab²`
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🔢 19. `-6ab - 5ab² - 4ab + 3ab²`
→ `(-6ab - 4ab) + (-5ab² + 3ab²)` = `-10ab - 2ab²`
✔ Answer: `-10ab - 2ab²`
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🔢 20. `-6a²b - 5ab² - 4a²b + 3ab²`
→ `(-6a²b - 4a²b) + (-5ab² + 3ab²)` = `-10a²b - 2ab²`
✔ Answer: `-10a²b - 2ab²`
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## 📝 Final Answers Summary:
1. `2a² + 2b²`
2. `7a² + 2b²`
3. `7a² + 6b²`
4. `10a² + 6b²`
5. `10a² + 8b²`
6. `2a² + 8b²`
7. `2a² + 2b²`
8. `2a² - 8b²`
9. `2a² - 8b`
10. `2a² - 8ab`
11. `2a² - 8ab`
12. `2a² - 8a²b`
13. `2a² - 2a²b`
14. `2a² - 5a²b + 3ab²`
15. `2a² - 2ab²`
16. `-10a² - 2ab²`
17. `-10 - 2ab²`
18. `-10a - 2ab²`
19. `-10ab - 2ab²`
20. `-10a²b - 2ab²`
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✔ All problems solved by identifying and combining like terms. Remember: only terms with identical variable parts can be combined!
Let me know if you want these explained in video format or need worksheets! 😊
Parent Tip: Review the logic above to help your child master the concept of collect like terms worksheet.
