You've uploaded a worksheet on the
Power Rule for exponents, and you’ve already filled in most of the answers by hand. Let’s go through each problem carefully, verify your answers, correct any mistakes, and explain the reasoning behind each solution.
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##
✔ The Power Rule:
>
(xᵃ)ᵇ = xᵃᵇ
This rule says: When you raise a power to another power, you multiply the exponents.
Also, remember:
-
Negative exponents: x⁻ⁿ = 1/xⁿ
-
Coefficient handling: (a·xᵇ)ᶜ = aᶜ · xᵇᶜ
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# 🔢 NUMERICAL BASES
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1. (8²)⁵
✔ Your answer:
8¹⁰ —
Correct!
→ Multiply exponents: 2 × 5 = 10 → 8¹⁰
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2. ((-15)⁶)³
✔ Your answer:
(-15)¹⁸ —
Correct!
→ Multiply exponents: 6 × 3 = 18 → (-15)¹⁸
*(Note: Even exponent on negative base → positive result, but we leave as is unless asked to evaluate numerically.)*
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3. (4⁻²)²
✘ Your answer:
4⁻⁴ = 1/4⁴ —
Technically correct, but let’s simplify fully.
→ Multiply exponents: -2 × 2 = -4 → 4⁻⁴
→ Convert to positive exponent:
1/4⁴ = 1/256
✔ Final Answer: 1/256
*(Your work shows you got 4⁻⁴ and then wrote 1/4⁴ — that’s good. But you should write the simplified fraction as 1/256.)*
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4. (2⁴)⁻²
✔ Your answer:
2⁻⁸ = 1/256 —
Correct!
→ Multiply exponents: 4 × (-2) = -8 → 2⁻⁸
→ Convert: 1/2⁸ = 1/256
*(You wrote “1/256” and circled it — perfect!)*
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5. (10⁻³)⁻⁷
✔ Your answer:
10²¹ —
Correct!
→ Multiply exponents: -3 × (-7) = 21 → 10²¹
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6. (3⁻¹)³
✔ Your answer:
3⁻³ = 1/27 —
Correct!
→ Multiply exponents: -1 × 3 = -3 → 3⁻³
→ Convert: 1/3³ = 1/27
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# 📐 VARIABLE BASES
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7. (x⁹)³
✔ Your answer:
x²⁷ —
Correct!
→ 9 × 3 = 27 → x²⁷
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8. (a²)⁷
✔ Your answer:
a¹⁴ —
Correct!
→ 2 × 7 = 14 → a¹⁴
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9. (k⁴)⁻⁵
✔ Your answer:
k⁻²⁰ = 1/k²⁰ —
Correct!
→ 4 × (-5) = -20 → k⁻²⁰ → 1/k²⁰
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10. (cd²)⁴
✔ Your answer:
c⁴d⁸ —
Correct!
→ Apply power to each factor: c¹ → c⁴, d² → d⁸
→ So: c⁴d⁸
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11. (m⁻²n⁸)³
✔ Your answer:
m⁻⁶n²⁴ = n²⁴/m⁶ —
Correct!
→ m⁻² → m⁻⁶, n⁸ → n²⁴
→ Write with positive exponents:
n²⁴ / m⁶
*(You wrote it as a fraction — excellent!)*
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12. (x⁻⁴y⁻²)⁻⁷
✔ Your answer:
x²⁸y¹⁴ —
Correct!
→ x⁻⁴ → x⁽⁻⁴⁾⁽⁻⁷⁾ = x²⁸
→ y⁻² → y⁽⁻²⁾⁽⁻⁷⁾ = y¹⁴
→ Final:
x²⁸y¹⁴
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# ➕ Examples with Coefficients
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13. (3x⁷)²
✔ Your answer:
3²x¹⁴ = 9x¹⁴ —
Correct!
→ Coefficient: 3² = 9
→ Variable: (x⁷)² = x¹⁴
→ Final:
9x¹⁴
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14. (5n⁴)³
✔ Your answer:
5³n¹² = 125n¹² —
Correct!
→ 5³ = 125, (n⁴)³ = n¹²
→ Final:
125n¹²
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15. (10x⁷)²
✘ Your answer:
10²x¹⁴ = 100x¹⁴ — You wrote “10x¹⁴”, which is
incorrect.
→ Coefficient: 10² =
100, not 10
→ Variable: (x⁷)² = x¹⁴
✔ Correct Answer: 100x¹⁴
*(You forgot to square the 10!)*
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16. (4w⁻⁷)⁴
✔ Your answer:
4⁴w⁻²⁸ = 256w⁻²⁸ = 256/w²⁸ —
Correct!
→ 4⁴ = 256
→ (w⁻⁷)⁴ = w⁻²⁸ → 1/w²⁸
→ Final:
256/w²⁸
*(You wrote “256/w²⁸” — perfect!)*
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## 📝 Summary of Corrections:
| Problem | Your Answer | Correct Answer | Notes |
|---------|-------------|----------------|-------|
| 3 | 4⁻⁴ = 1/4⁴ |
1/256 | Simplify 4⁴ to 256 |
| 15 | 10x¹⁴ |
100x¹⁴ | Forgot to square the coefficient 10 |
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##
✔ Final Grade:
A- (Mostly perfect — just two small errors!)
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## 💡 Tips for Next Time:
- Always
square the entire coefficient, not just the variable.
- When writing negative exponents as fractions,
simplify the denominator (e.g., 4⁴ = 256, not left as 4⁴).
- Double-check signs when multiplying negative exponents.
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Let me know if you’d like to practice more problems or review the rules again! 😊
Parent Tip: Review the logic above to help your child master the concept of powers of monomials worksheet.
