Exponents Worksheets with Answer Key - Free Printable

Let’s solve each problem step by step using exponent rules. I’ll go one by one, check my work, and then give the final answers.

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1. (x⁴)²
→ Power of a power: multiply exponents → 4 × 2 = 8
→ Answer: x⁸

2. x⁻⁸
→ Already simplified. Negative exponent means reciprocal, but since no denominator is given, we leave as is unless asked to rewrite.
→ But “simplify” usually means write with positive exponents if possible — however, this is already in simplest form for an expression like this.
Wait — actually, looking at context, maybe they just want it left as is? Let me check others.
Actually, in many worksheets, x⁻⁸ is considered simplified unless told to write without negative exponents. But let’s see #7 and #13 — those have negative exponents too. So probably acceptable.
But wait — instruction says “Simplify”. In most middle/high school contexts, simplifying includes writing with positive exponents only if it’s part of a fraction or product. Here, it’s standalone.
I think we leave it as x⁻⁸, but let me double-check standard practice. Actually, some teachers expect you to write 1/x⁸. Hmm.
Looking at #15: -(9a)⁰ — that’s -1, so they’re testing zero exponent.
#16: 1/2⁻⁶ — that will become 2⁶. So yes, they likely want positive exponents where possible.
So for #2: x⁻⁸ = 1/x⁸

But wait — let’s hold off and do all first, then decide based on pattern.

Actually, let’s follow strict rule: “Simplify” means apply exponent rules and write in simplest form, which often means no negative exponents in final answer unless specified otherwise.

I’ll assume we convert negative exponents to fractions.

So:

2. x⁻⁸ = 1/x⁸

3. (a²)⁰
→ Anything (except 0) to the 0 power is 1.
→ a² is not zero (assuming a ≠ 0), so answer is 1

4. 2a² · 3b
→ Multiply coefficients: 2 × 3 = 6
→ Variables: a² and b — different bases, so just write together
→ Answer: 6a²b

5. (4x²)⁻⁴
→ Apply exponent to both coefficient and variable: 4⁻⁴ · (x²)⁻⁴
→ 4⁻⁴ = 1/4⁴ = 1/256
→ (x²)⁻⁴ = x⁻⁸ = 1/x⁸
→ So overall: 1/(256x⁸)
→ Answer: 1/(256x⁸)

6. (4a⁴)²
→ Square the coefficient: 4² = 16
→ Square the variable: (a⁴)² = a⁸
→ Answer: 16a⁸

7. (4ab)⁻¹
→ Negative exponent means reciprocal
→ So: 1/(4ab)
→ Answer: 1/(4ab)

8. (a²b⁻¹)²
→ Apply exponent to each factor inside: (a²)² · (b⁻¹)² = a⁴ · b²
→ Write with positive exponent: a⁴ / b²
→ Answer: a⁴/b²

9. (6ab)²
→ Square each part: 6² = 36, a², b²
→ Answer: 36a²b²

10. 18a³ / 4a
→ Simplify coefficients: 18/4 = 9/2
→ Simplify variables: a³/a = a²
→ Answer: (9/2)a² or 9a²/2

11. 2a³ / a²
→ Coefficient stays 2
→ a³/a² = a¹ = a
→ Answer: 2a

12. (3a²b⁷ / a)⁵
→ First simplify inside parentheses:
- Coefficient: 3 remains
- a² / a = a¹ = a
- b⁷ remains
→ So inside: 3a b⁷
→ Now raise to 5th power: (3ab⁷)⁵ = 3⁵ · a⁵ · (b⁷)⁵ = 243 · a⁵ · b³⁵
→ Answer: 243a⁵b³⁵

13. a⁻¹ / a⁻⁸
→ Use quotient rule: subtract exponents → (-1) - (-8) = -1 + 8 = 7
→ So a⁷
Alternatively: (1/a) / (1/a⁸) = (1/a) * (a⁸/1) = a⁷
→ Answer: a⁷

14. x⁵y⁴ / xy³
→ x⁵/x = x⁴
→ y⁴/y³ = y¹ = y
→ Answer: x⁴y

15. -(9a)⁰
→ (9a)⁰ = 1 (as long as 9a ≠ 0, which we assume)
→ Then apply the negative sign: -1
→ Answer: -1

16. 1 / 2⁻⁶
→ Dividing by a negative exponent is same as multiplying by positive: 1 * 2⁶ = 64
→ Or: 1 / (1/2⁶) = 2⁶ = 64
→ Answer: 64

17. a⁸ · a⁻⁷
→ Product rule: add exponents → 8 + (-7) = 1
→ So a¹ = a
→ Answer: a

18. (a²b)⁴
→ Apply exponent to each: (a²)⁴ = a⁸, b⁴
→ Answer: a⁸b⁴

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Now, going back to #2: x⁻⁸. Since other problems like #7, #8, #13 involve converting negative exponents to fractions, and #16 converts denominator negative exponent to whole number, it’s consistent to write #2 as 1/x⁸.

Also, for #10, I wrote 9a²/2 — that’s fine.

Let me verify a few tricky ones again:

- #5: (4x²)⁻⁴ = 4⁻⁴ x⁻⁸ = 1/(256x⁸) ✔
- #12: Inside: 3a²b⁷ / a = 3a b⁷ → raised to 5: 243 a⁵ b³⁵ ✔
- #13: a⁻¹ / a⁻⁸ = a^{(-1)-(-8)} = a⁷ ✔
- #15: -(9a)^0 = -1 ✔ (note: the negative is outside the parentheses)

All look good.

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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