Collecting like terms (with powers) - Variation Theory - Free Printable

Let's solve each of these algebraic expressions by combining like terms. Like terms are terms that have the same variables raised to the same powers.

We will go through each problem one by one and simplify.

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1. $ a^2 + b^2 + a^2 + b^2 $

Group like terms:
- $ a^2 + a^2 = 2a^2 $
- $ b^2 + b^2 = 2b^2 $

✔ Answer: $ 2a^2 + 2b^2 $

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2. $ 6a^2 + b^2 + a^2 + b^2 $

Group:
- $ 6a^2 + a^2 = 7a^2 $
- $ b^2 + b^2 = 2b^2 $

✔ Answer: $ 7a^2 + 2b^2 $

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3. $ 6a^2 + 5b^2 + a^2 + b^2 $

Group:
- $ 6a^2 + a^2 = 7a^2 $
- $ 5b^2 + b^2 = 6b^2 $

✔ Answer: $ 7a^2 + 6b^2 $

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4. $ 6a^2 + 5b^2 + 4a^2 + b^2 $

Group:
- $ 6a^2 + 4a^2 = 10a^2 $
- $ 5b^2 + b^2 = 6b^2 $

✔ Answer: $ 10a^2 + 6b^2 $

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5. $ 6a^2 + 5b^2 + 4a^2 + 3b^2 $

Group:
- $ 6a^2 + 4a^2 = 10a^2 $
- $ 5b^2 + 3b^2 = 8b^2 $

✔ Answer: $ 10a^2 + 8b^2 $

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6. $ 6a^2 + 5b^2 - 4a^2 + 3b^2 $

Group:
- $ 6a^2 - 4a^2 = 2a^2 $
- $ 5b^2 + 3b^2 = 8b^2 $

✔ Answer: $ 2a^2 + 8b^2 $

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7. $ 6a^2 + 5b^2 - 4a^2 - 3b^2 $

Group:
- $ 6a^2 - 4a^2 = 2a^2 $
- $ 5b^2 - 3b^2 = 2b^2 $

✔ Answer: $ 2a^2 + 2b^2 $

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8. $ 6a^2 - 5b^2 - 4a^2 - 3b^2 $

Group:
- $ 6a^2 - 4a^2 = 2a^2 $
- $ -5b^2 - 3b^2 = -8b^2 $

✔ Answer: $ 2a^2 - 8b^2 $

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9. $ 6a^2 - 5b - 4a^2 - 3b $

Group:
- $ 6a^2 - 4a^2 = 2a^2 $
- $ -5b - 3b = -8b $

✔ Answer: $ 2a^2 - 8b $

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10. $ 6a^2 - 5ab - 4a^2 - 3ab $

Group:
- $ 6a^2 - 4a^2 = 2a^2 $
- $ -5ab - 3ab = -8ab $

✔ Answer: $ 2a^2 - 8ab $

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11. $ 6a^2 - 5ab - 4a^2 - 3ba $

Note: $ ab = ba $, so $ -3ba = -3ab $

So:
- $ 6a^2 - 4a^2 = 2a^2 $
- $ -5ab - 3ab = -8ab $

✔ Answer: $ 2a^2 - 8ab $

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12. $ 6a^2 - 5a^2b - 4a^2 - 3a^2b $

Group:
- $ 6a^2 - 4a^2 = 2a^2 $
- $ -5a^2b - 3a^2b = -8a^2b $

✔ Answer: $ 2a^2 - 8a^2b $

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13. $ 6a^2 - 5a^2b - 4a^2 + 3a^2b $

Group:
- $ 6a^2 - 4a^2 = 2a^2 $
- $ -5a^2b + 3a^2b = -2a^2b $

✔ Answer: $ 2a^2 - 2a^2b $

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14. $ 6a^2 - 5a^2b - 4a^2 + 3ab^2 $

Group:
- $ 6a^2 - 4a^2 = 2a^2 $
- $ -5a^2b $ (no other $ a^2b $ term)
- $ +3ab^2 $ (no other $ ab^2 $)

So:
- $ 2a^2 $
- $ -5a^2b $
- $ +3ab^2 $

✔ Answer: $ 2a^2 - 5a^2b + 3ab^2 $

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15. $ 6a^2 - 5ab^2 - 4a^2 + 3ab^2 $

Group:
- $ 6a^2 - 4a^2 = 2a^2 $
- $ -5ab^2 + 3ab^2 = -2ab^2 $

✔ Answer: $ 2a^2 - 2ab^2 $

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16. $ -6a^2 - 5ab^2 - 4a^2 + 3ab^2 $

Group:
- $ -6a^2 - 4a^2 = -10a^2 $
- $ -5ab^2 + 3ab^2 = -2ab^2 $

✔ Answer: $ -10a^2 - 2ab^2 $

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17. $ -6 - 5ab^2 - 4 + 3ab^2 $

Group constants:
- $ -6 - 4 = -10 $
- $ -5ab^2 + 3ab^2 = -2ab^2 $

✔ Answer: $ -10 - 2ab^2 $

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18. $ -6a - 5ab^2 - 4a + 3ab^2 $

Group:
- $ -6a - 4a = -10a $
- $ -5ab^2 + 3ab^2 = -2ab^2 $

✔ Answer: $ -10a - 2ab^2 $

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19. $ -6ab - 5ab^2 - 4ab + 3ab^2 $

Group:
- $ -6ab - 4ab = -10ab $
- $ -5ab^2 + 3ab^2 = -2ab^2 $

✔ Answer: $ -10ab - 2ab^2 $

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20. $ -6a^2b - 5ab^2 - 4a^2b + 3ab^2 $

Group:
- $ -6a^2b - 4a^2b = -10a^2b $
- $ -5ab^2 + 3ab^2 = -2ab^2 $

✔ Answer: $ -10a^2b - 2ab^2 $

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✔ Final Answers:

| Problem | Simplified Expression |
|--------|------------------------|
| 1 | $ 2a^2 + 2b^2 $ |
| 2 | $ 7a^2 + 2b^2 $ |
| 3 | $ 7a^2 + 6b^2 $ |
| 4 | $ 10a^2 + 6b^2 $ |
| 5 | $ 10a^2 + 8b^2 $ |
| 6 | $ 2a^2 + 8b^2 $ |
| 7 | $ 2a^2 + 2b^2 $ |
| 8 | $ 2a^2 - 8b^2 $ |
| 9 | $ 2a^2 - 8b $ |
| 10 | $ 2a^2 - 8ab $ |
| 11 | $ 2a^2 - 8ab $ |
| 12 | $ 2a^2 - 8a^2b $ |
| 13 | $ 2a^2 - 2a^2b $ |
| 14 | $ 2a^2 - 5a^2b + 3ab^2 $ |
| 15 | $ 2a^2 - 2ab^2 $ |
| 16 | $ -10a^2 - 2ab^2 $ |
| 17 | $ -10 - 2ab^2 $ |
| 18 | $ -10a - 2ab^2 $ |
| 19 | $ -10ab - 2ab^2 $ |
| 20 | $ -10a^2b - 2ab^2 $ |

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🔍 Key Concepts Used:

- Combining like terms: Terms with the same variables and exponents.
- Commutative property: Order doesn’t matter in addition.
- Alphabetical order: Usually written as $ a $ before $ b $, but not required unless specified.
- Note: $ ab = ba $, so $ -3ba = -3ab $

Let me know if you'd like these arranged neatly in a printable format or explained further!