Multiplying Exponents With Different Bases and the Same Exponent ... - Free Printable
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Step-by-step solution for: Multiplying Exponents With Different Bases and the Same Exponent ...
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Show Answer Key & Explanations
Step-by-step solution for: Multiplying Exponents With Different Bases and the Same Exponent ...
Actually, the solutions provided in the image are mostly incorrect. The worksheet is titled “Multiplying Exponents (A) Answers,” but the method used — combining the bases first and then applying the exponent — is only valid when the exponents are the same, which is not the case here.
---
## 🔍 Let’s clarify the correct rule:
When multiplying two exponential expressions with the same exponent, you can combine the bases:
> aⁿ · bⁿ = (a·b)ⁿ
✔ This rule works because:
> aⁿ · bⁿ = (a·a·...·a) · (b·b·...·b) = (a·b)·(a·b)·...·(a·b) = (a·b)ⁿ
---
But if the exponents are different, you cannot combine the bases like that.
For example:
> 5⁻⁵ · (-2)⁻⁵ → same exponent → ✔ (5·-2)⁻⁵ = (-10)⁻⁵ → Correct!
But:
> 4⁻⁸ · 8⁻⁸ → same exponent → ✔ (4·8)⁻⁸ = 32⁻⁸ → Correct!
Wait — actually, looking again, all problems in this worksheet have the same exponent on both terms!
Let’s check each one:
1. 5⁻⁵ · (-2)⁻⁵ → both exponent -5 → ✔
2. (-3)⁻² · (-9)⁻² → both exponent -2 → ✔
3. 4⁻⁸ · 8⁻⁸ → both exponent -8 → ✔
4. 2⁻¹ · 8⁻¹ → both exponent -1 → ✔
5. 3⁷ · (-7)⁷ → both exponent 7 → ✔
6. (-7)⁻² · (-5)⁻² → both exponent -2 → ✔
7. 7³ · (-6)³ → both exponent 3 → ✔
8. 6⁵ · 9⁵ → both exponent 5 → ✔
9. (-9)⁻⁹ · 4⁻⁹ → both exponent -9 → ✔
10. (-4)⁻⁹ · 8⁻⁹ → both exponent -9 → ✔
---
## 🎯 So the rule applies to ALL of them!
The method shown — multiplying the bases first and keeping the exponent — is correct for all these problems.
BUT — there’s a catch: some of the final simplifications are wrong or misleading.
---
## ✔ Let’s go through each problem correctly:
---
= [5 × (-2)]⁻⁵ = (-10)⁻⁵ = 1 / (-10)⁵ = 1 / (-100000) = -1/100000
✘ The answer sheet says: `(-10)⁻⁵ = 1 / (-10)⁵` — technically true, but it should be simplified to -1/100000, not left as 1/(-10)⁵ unless specified.
---
= [(-3) × (-9)]⁻² = (27)⁻² = 1 / 27² = 1/729
✔ Correctly simplified.
---
= (4 × 8)⁻⁸ = 32⁻⁸ = 1 / 32⁸
✔ Correct.
---
= (2 × 8)⁻¹ = 16⁻¹ = 1/16
✔ Correct.
---
= [3 × (-7)]⁷ = (-21)⁷
Now, (-21)⁷ is a negative number because 7 is odd.
✔ The answer sheet shows `(-21)⁷` — correct form, though it could be written as -21⁷ for clarity (but mathematically equivalent).
---
= [(-7) × (-5)]⁻² = (35)⁻² = 1 / 35² = 1/1225
✔ Correct.
---
= [7 × (-6)]³ = (-42)³ = -74088
✔ Answer sheet says `(-42)³` — acceptable, but if simplifying fully, it’s -74088.
---
= (6 × 9)⁵ = 54⁵
✔ Correct. 54⁵ is fine as a final answer unless asked to compute numerically.
---
= [(-9) × 4]⁻⁹ = (-36)⁻⁹ = 1 / (-36)⁹
Since 9 is odd, (-36)⁹ is negative → so this equals -1 / 36⁹
✘ The answer sheet says `1 / (-36)⁹` — which is mathematically correct, but it’s equal to -1/36⁹, so it’s better to write it as such for clarity.
---
= [(-4) × 8]⁻⁹ = (-32)⁻⁹ = 1 / (-32)⁹ = -1 / 32⁹
✘ Same issue — answer sheet says `1 / (-32)⁹`, which is correct but should be simplified to -1/32⁹.
---
## ✔ Final Summary:
- The method used (combining bases when exponents are equal) is mathematically correct.
- The answers are mostly correct, but some need sign correction when the base is negative and the exponent is odd.
- Specifically, Problems 1, 9, and 10 should reflect the negative sign in the final simplified fraction.
---
## ✏️ Corrected Final Answers:
1. -1/100000
2. 1/729
3. 1/32⁸
4. 1/16
5. (-21)⁷ or -21⁷
6. 1/1225
7. (-42)³ or -74088
8. 54⁵
9. -1/36⁹
10. -1/32⁹
---
## 🧠 Key Takeaway:
> When multiplying two numbers with the same exponent, you can multiply the bases and keep the exponent:
> aⁿ · bⁿ = (a·b)ⁿ
Just remember: if the result inside the parentheses is negative and the exponent is odd, the final result is negative. If the exponent is even, the result is positive.
---
✔ So while the worksheet’s *method* is correct, its *final simplified forms* for problems with negative bases and odd exponents are incomplete or misleading. The answers should reflect the correct sign.
Let me know if you’d like to see numerical values or step-by-step breakdowns for any specific problem!
---
## 🔍 Let’s clarify the correct rule:
When multiplying two exponential expressions with the same exponent, you can combine the bases:
> aⁿ · bⁿ = (a·b)ⁿ
✔ This rule works because:
> aⁿ · bⁿ = (a·a·...·a) · (b·b·...·b) = (a·b)·(a·b)·...·(a·b) = (a·b)ⁿ
---
But if the exponents are different, you cannot combine the bases like that.
For example:
> 5⁻⁵ · (-2)⁻⁵ → same exponent → ✔ (5·-2)⁻⁵ = (-10)⁻⁵ → Correct!
But:
> 4⁻⁸ · 8⁻⁸ → same exponent → ✔ (4·8)⁻⁸ = 32⁻⁸ → Correct!
Wait — actually, looking again, all problems in this worksheet have the same exponent on both terms!
Let’s check each one:
1. 5⁻⁵ · (-2)⁻⁵ → both exponent -5 → ✔
2. (-3)⁻² · (-9)⁻² → both exponent -2 → ✔
3. 4⁻⁸ · 8⁻⁸ → both exponent -8 → ✔
4. 2⁻¹ · 8⁻¹ → both exponent -1 → ✔
5. 3⁷ · (-7)⁷ → both exponent 7 → ✔
6. (-7)⁻² · (-5)⁻² → both exponent -2 → ✔
7. 7³ · (-6)³ → both exponent 3 → ✔
8. 6⁵ · 9⁵ → both exponent 5 → ✔
9. (-9)⁻⁹ · 4⁻⁹ → both exponent -9 → ✔
10. (-4)⁻⁹ · 8⁻⁹ → both exponent -9 → ✔
---
## 🎯 So the rule applies to ALL of them!
The method shown — multiplying the bases first and keeping the exponent — is correct for all these problems.
BUT — there’s a catch: some of the final simplifications are wrong or misleading.
---
## ✔ Let’s go through each problem correctly:
---
1. 5⁻⁵ · (-2)⁻⁵
= [5 × (-2)]⁻⁵ = (-10)⁻⁵ = 1 / (-10)⁵ = 1 / (-100000) = -1/100000
✘ The answer sheet says: `(-10)⁻⁵ = 1 / (-10)⁵` — technically true, but it should be simplified to -1/100000, not left as 1/(-10)⁵ unless specified.
---
2. (-3)⁻² · (-9)⁻²
= [(-3) × (-9)]⁻² = (27)⁻² = 1 / 27² = 1/729
✔ Correctly simplified.
---
3. 4⁻⁸ · 8⁻⁸
= (4 × 8)⁻⁸ = 32⁻⁸ = 1 / 32⁸
✔ Correct.
---
4. 2⁻¹ · 8⁻¹
= (2 × 8)⁻¹ = 16⁻¹ = 1/16
✔ Correct.
---
5. 3⁷ · (-7)⁷
= [3 × (-7)]⁷ = (-21)⁷
Now, (-21)⁷ is a negative number because 7 is odd.
✔ The answer sheet shows `(-21)⁷` — correct form, though it could be written as -21⁷ for clarity (but mathematically equivalent).
---
6. (-7)⁻² · (-5)⁻²
= [(-7) × (-5)]⁻² = (35)⁻² = 1 / 35² = 1/1225
✔ Correct.
---
7. 7³ · (-6)³
= [7 × (-6)]³ = (-42)³ = -74088
✔ Answer sheet says `(-42)³` — acceptable, but if simplifying fully, it’s -74088.
---
8. 6⁵ · 9⁵
= (6 × 9)⁵ = 54⁵
✔ Correct. 54⁵ is fine as a final answer unless asked to compute numerically.
---
9. (-9)⁻⁹ · 4⁻⁹
= [(-9) × 4]⁻⁹ = (-36)⁻⁹ = 1 / (-36)⁹
Since 9 is odd, (-36)⁹ is negative → so this equals -1 / 36⁹
✘ The answer sheet says `1 / (-36)⁹` — which is mathematically correct, but it’s equal to -1/36⁹, so it’s better to write it as such for clarity.
---
10. (-4)⁻⁹ · 8⁻⁹
= [(-4) × 8]⁻⁹ = (-32)⁻⁹ = 1 / (-32)⁹ = -1 / 32⁹
✘ Same issue — answer sheet says `1 / (-32)⁹`, which is correct but should be simplified to -1/32⁹.
---
## ✔ Final Summary:
- The method used (combining bases when exponents are equal) is mathematically correct.
- The answers are mostly correct, but some need sign correction when the base is negative and the exponent is odd.
- Specifically, Problems 1, 9, and 10 should reflect the negative sign in the final simplified fraction.
---
## ✏️ Corrected Final Answers:
1. -1/100000
2. 1/729
3. 1/32⁸
4. 1/16
5. (-21)⁷ or -21⁷
6. 1/1225
7. (-42)³ or -74088
8. 54⁵
9. -1/36⁹
10. -1/32⁹
---
## 🧠 Key Takeaway:
> When multiplying two numbers with the same exponent, you can multiply the bases and keep the exponent:
> aⁿ · bⁿ = (a·b)ⁿ
Just remember: if the result inside the parentheses is negative and the exponent is odd, the final result is negative. If the exponent is even, the result is positive.
---
✔ So while the worksheet’s *method* is correct, its *final simplified forms* for problems with negative bases and odd exponents are incomplete or misleading. The answers should reflect the correct sign.
Let me know if you’d like to see numerical values or step-by-step breakdowns for any specific problem!
Parent Tip: Review the logic above to help your child master the concept of multiplying powers with the same base worksheet.
