Let’s work through each problem step by step. We’re simplifying expressions using exponent rules, and we must make sure all exponents in the final answer are positive.
---
Problem 1:
2m² · 2m³
→ Multiply coefficients: 2 × 2 = 4
→ Add exponents of same base: m²⁺³ = m⁵
✔ Final:
4m⁵
---
Problem 2:
m⁴ · 2m⁻³
→ Coefficient: 1 × 2 = 2
→ Exponents: m⁴⁺⁽⁻³⁾ = m¹ → m
✔ Final:
2m
---
Problem 3:
4r⁻³ · 2r²
→ Coefficients: 4 × 2 = 8
→ Exponents: r⁻³⁺² = r⁻¹ → write as 1/r
✔ Final:
8/r
---
Problem 4:
4n⁴ · 2n⁻³
→ Coefficients: 4 × 2 = 8
→ Exponents: n⁴⁺⁽⁻³⁾ = n¹ → n
✔ Final:
8n
---
Problem 5:
2k⁴ · 4k
→ Coefficients: 2 × 4 = 8
→ k⁴ · k¹ = k⁵
✔ Final:
8k⁵
---
Problem 6:
2x³y⁻³ · 2x⁻¹y³
→ Coefficients: 2 × 2 = 4
→ x: x³⁺⁽⁻¹⁾ = x²
→ y: y⁻³⁺³ = y⁰ = 1
✔ Final:
4x²
---
Problem 7:
2y² · 3x
→ Coefficients: 2 × 3 = 6
→ Variables: y² and x (no like bases to combine)
✔ Final:
6y²x or
6xy² — but looking at answer options, “6y²x” is listed.
Wait — let’s check the answer options again. The last option is “6y²x” — yes, that matches.
✔ Final:
6y²x
---
Problem 8:
4v³ · vu²
→ Write v as v¹ → v³ · v¹ = v⁴
→ u² stays
→ Coefficient: 4
✔ Final:
4v⁴u²
---
Problem 9:
4a³b² · 3a⁻⁴b⁻³
→ Coefficients: 4 × 3 = 12
→ a: a³⁺⁽⁻⁴⁾ = a⁻¹ → 1/a
→ b: b²⁺⁽⁻³⁾ = b⁻¹ → 1/b
→ So: 12 / (a·b) = 12/(ab)
✔ Final:
12/(ab)
---
Problem 10:
x²y⁻⁴ · x³y²
→ x: x²⁺³ = x⁵
→ y: y⁻⁴⁺² = y⁻² → 1/y²
✔ Final:
x⁵/y²
---
Problem 11:
(x²)⁰
→ Anything to the 0 power is 1 (as long as it’s not zero)
✔ Final:
1
---
Problem 12:
(2x²)⁻⁴
→ Apply exponent to both 2 and x²: 2⁻⁴ · (x²)⁻⁴ = 1/16 · x⁻⁸ → 1/(16x⁸)
✔ Final:
1/(16x⁸)
---
Problem 13:
(4r⁰)⁴
→ r⁰ = 1 → 4·1 = 4 → 4⁴ = 256
✔ Final:
256
---
Problem 14:
(4a³)²
→ Square both: 4² = 16, (a³)² = a⁶
✔ Final:
16a⁶
---
Problem 15:
(3k⁴)⁴
→ 3⁴ = 81, (k⁴)⁴ = k¹⁶
✔ Final:
81k¹⁶
---
Problem 16:
(4xy)⁻¹
→ Negative exponent means reciprocal: 1/(4xy)
✔ Final:
1/(4xy)
---
Now, let’s match our answers to the “Answer Options” on the right side of the worksheet:
Answer Options listed:
- 12/(ab) → matches #9
- 1 → matches #11
- 8n → matches #4
- 2m → matches #2
- 8/r → matches #3
- 4x² → matches #6
- 16a⁶ → matches #14
- 1/(4xy) → matches #16
- 4m⁵ → matches #1
- x⁵/y² → matches #10
- 256 → matches #13
- 8k⁵ → matches #5
- 81k¹⁶ → matches #15
- 1/(16x⁸) → matches #12
- 4v⁴u² → matches #8
- 6y²x → matches #7
Perfect! All problems have matching answers in the list.
Final Answer:
1) 4m⁵
2) 2m
3) 8/r
4) 8n
5) 8k⁵
6) 4x²
7) 6y²x
8) 4v⁴u²
9) 12/(ab)
10) x⁵/y²
11) 1
12) 1/(16x⁸)
13) 256
14) 16a⁶
15) 81k¹⁶
16) 1/(4xy)
Parent Tip: Review the logic above to help your child master the concept of properties of exponents worksheet.
