Let’s solve each problem one by one, using the rules of exponents. I’ll go step by step for each.
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1. (x⁴)²
When you raise a power to another power, multiply the exponents:
(x⁴)² = x^(4×2) =
x⁸
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2. x⁻⁸
Negative exponent means “flip” it to make it positive:
x⁻⁸ =
1/x⁸
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3. (a²)⁰
Anything (except 0) raised to the 0 power is 1:
(a²)⁰ =
1
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4. 2a² · 3b
Multiply numbers and keep variables separate:
2 × 3 = 6 → so we get
6a²b
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5. (4x²)⁻⁴
First, apply the exponent to both 4 and x²:
= 4⁻⁴ · (x²)⁻⁴
= 1/4⁴ · x^(-8)
4⁴ = 256 → so this is
1/(256x⁸)
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6. (4a⁴)²
Apply exponent to both 4 and a⁴:
= 4² · (a⁴)² = 16 · a⁸ =
16a⁸
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7. (4ab)⁻¹
Negative exponent means flip the whole thing:
= 1/(4ab) →
1/(4ab)
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8. (a²b⁻¹)²
Apply exponent to each part inside:
= (a²)² · (b⁻¹)² = a⁴ · b⁻²
Now write with positive exponents:
a⁴ / b²
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9. (6ab)²
Square each part:
= 6² · a² · b² =
36a²b²
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10. 18a³ / 4a
Simplify numbers: 18/4 = 9/2
Simplify variables: a³/a = a²
So:
(9/2)a² or
9a²/2
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11. 2a³ / a²
Numbers stay as 2. Variables: a³/a² = a¹ = a
Answer:
2a
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12. (3a²b⁷ / a)⁵
First simplify inside the parentheses:
a² / a = a¹ → so inside becomes 3a b⁷
Now raise to 5th power:
(3a b⁷)⁵ = 3⁵ · a⁵ · (b⁷)⁵ = 243 · a⁵ · b³⁵
Answer:
243a⁵b³⁵
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13. a⁻¹ / a⁻⁸
Use quotient rule: subtract exponents when dividing same base:
a^(-1 - (-8)) = a^(-1 + 8) = a⁷
Answer:
a⁷
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14. x⁵y⁴ / xy³
Break into parts:
x⁵/x = x⁴
y⁴/y³ = y¹ = y
Answer:
x⁴y
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15. -(9a)⁰
Anything to the 0 power is 1 → (9a)⁰ = 1
Then apply the negative sign:
-1
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16. 1 / 2⁻⁶
Negative exponent in denominator? Flip it up:
1 / 2⁻⁶ = 2⁶ =
64
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17. a⁸ · a⁻⁷
When multiplying same base, add exponents:
a^(8 + (-7)) = a¹ =
a
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18. (a²b)⁴
Apply exponent to each part:
(a²)⁴ · b⁴ = a⁸ · b⁴ =
a⁸b⁴
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Final Answer:
1. x⁸
2. 1/x⁸
3. 1
4. 6a²b
5. 1/(256x⁸)
6. 16a⁸
7. 1/(4ab)
8. a⁴/b²
9. 36a²b²
10. 9a²/2
11. 2a
12. 243a⁵b³⁵
13. a⁷
14. x⁴y
15. -1
16. 64
17. a
18. a⁸b⁴
Parent Tip: Review the logic above to help your child master the concept of exponent rules worksheet with answers.
