Let's solve each problem step by step, simplifying using the
properties of exponents, and ensuring all final answers have
only positive exponents.
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1) 2m² · 2m³
Multiply coefficients: 2 × 2 = 4
Add exponents of same base: m² · m³ = m^(2+3) = m⁵
✔ Answer: 4m⁵
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2) m⁴ · 2m⁻³
Multiply coefficients: 1 × 2 = 2 (since m⁴ has implied coefficient 1)
Add exponents: m⁴ · m⁻³ = m^(4-3) = m¹ = m
✔ Answer: 2m
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3) 4r⁻³ · 2r²
Coefficients: 4 × 2 = 8
Exponents: r⁻³ · r² = r^(-3+2) = r⁻¹ → convert to positive exponent: 1/r
✔ Answer: 8/r
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4) 4n⁴ · 2n⁻³
Coefficients: 4 × 2 = 8
Exponents: n⁴ · n⁻³ = n^(4-3) = n¹ = n
✔ Answer: 8n
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5) 2k⁴ · 4k
Coefficients: 2 × 4 = 8
Exponents: k⁴ · k¹ = k⁵
✔ Answer: 8k⁵
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6) 2x³y⁻³ · 2x⁻¹y³
Coefficients: 2 × 2 = 4
x exponents: x³ · x⁻¹ = x^(3-1) = x²
y exponents: y⁻³ · y³ = y⁰ = 1
So: 4x² · 1 = 4x²
✔ Answer: 4x²
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7) 2y² · 3x
No common bases → just multiply coefficients and write variables in order.
2 × 3 = 6, so 6xy² (alphabetical order preferred)
✔ Answer: 6y²x (or 6xy² — both are correct; worksheet uses 6y²x)
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8) 4v³ · vu²
Note: v is same as v¹ → so v³ · v¹ = v⁴
u² stays as is.
Coefficient: 4
✔ Answer: 4v⁴u²
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9) 4a³b² · 3a⁻⁴b⁻³
Coefficients: 4 × 3 = 12
a exponents: a³ · a⁻⁴ = a^(-1) = 1/a
b exponents: b² · b⁻³ = b^(-1) = 1/b
So: 12 / (ab)
✔ Answer: 12/(ab)
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10) x²y⁻⁴ · x³y²
x exponents: x² · x³ = x⁵
y exponents: y⁻⁴ · y² = y^(-2) = 1/y²
So: x⁵ / y²
✔ Answer: x⁵/y²
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11) (x²)⁰
Any non-zero expression to the power 0 is 1.
✔ Answer: 1
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12) (2x²)⁻⁴
Apply exponent to both coefficient and variable:
= 2⁻⁴ · (x²)⁻⁴ = (1/2⁴) · x^(-8) = (1/16) · (1/x⁸) = 1/(16x⁸)
✔ Answer: 1/(16x⁸)
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13) (4r⁰)⁴
r⁰ = 1 → so (4·1)⁴ = 4⁴ = 256
✔ Answer: 256
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14) (4a³)²
= 4² · (a³)² = 16 · a⁶ = 16a⁶
✔ Answer: 16a⁶
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15) (3k⁴)⁴
= 3⁴ · (k⁴)⁴ = 81 · k¹⁶ = 81k¹⁶
✔ Answer: 81k¹⁶
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16) (4xy)⁻¹
= 4⁻¹ · x⁻¹ · y⁻¹ = (1/4) · (1/x) · (1/y) = 1/(4xy)
✔ Answer: 1/(4xy)
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##
✔ Final Answer Key:
| Problem | 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) |
You can now match these with the
Answer Options on the right side of your worksheet. All answers are simplified with
positive exponents only, as required.
Let me know if you’d like to see any step explained further!
Parent Tip: Review the logic above to help your child master the concept of exponent properties worksheet.
