Exponents Worksheets with Answer Key - Free Printable

Let’s solve each problem step by step using the laws of exponents. We’ll make sure all answers have positive exponents.

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Problem ①: (a⁴)²

When you raise a power to another power, multiply the exponents:
(a⁴)² = a^(4×2) = a⁸

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Problem ②: p⁶ · p¹⁴

When multiplying same bases, add the exponents:
p⁶ · p¹⁴ = p^(6+14) = p²⁰

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Problem ③: p / p⁵

When dividing same bases, subtract the exponents:
p⁷ / p⁵ = p^(7-5) = p²

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Problem ④: (z²)³

Power to a power → multiply exponents:
(z²)³ = z^(2×3) = z⁶

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Problem ⑤: q¹⁰ / q⁶

Divide → subtract exponents:
q¹⁰ / q = q^(10-6) = q⁴

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Problem ⑥: l² / l

l is the same as l¹, so:
l² / l¹ = l^(2-1) = l¹ = l

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Problem ⑦: (x³b)⁴(xb⁶)²

First, apply exponent to each part inside parentheses:

→ (x³b)⁴ = x^(3×4) · b⁴ = x¹²b⁴
→ (xb⁶)² = x² · b^(6×2) = x²b¹²

Now multiply them together:
x¹²b⁴ · x²b¹² = x^(12+2) · b^(4+12) = x¹⁴b¹⁶

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Problem ⑧: (a²b / b⁻³c⁴)³ · (a⁻³b)⁻²

Break it into two parts.

Part A: (a²b / b⁻³c⁴)³

First simplify inside the parentheses:

Numerator: a²b
Denominator: b⁻³c → move b⁻³ to numerator as b³ (since negative exponent flips fraction)

So: a²b · b³ / c⁴ = a²b⁴ / c⁴

Now raise to power 3:
(a²b⁴ / c)³ = a^(2×3) b^(4×3) / c^(4×3) = a⁶b¹² / c¹²

Part B: (a⁻³b)⁻²

Apply exponent -2 to both terms:
= a^(-3 × -2) · b^(-2) = a⁶ · b⁻² = a⁶ / b²

Now multiply Part A and Part B:

(a⁶b¹² / c¹²) · (a⁶ / b²) = a^(6+6) · b^(12-2) / c¹² = a¹²b¹⁰ / c¹²

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Problem ⑨: (x²y⁻⁷ / x²y⁴)²

Simplify inside first:

x² / x⁻² = x^(2 - (-2)) = x⁴
y⁷ / y⁴ = y^(-7 - 4) = y⁻¹¹

So inside becomes: x⁴y⁻¹¹

Now square it:
(x⁴y⁻¹¹)² = x⁸y⁻²²

Make exponent positive: y⁻²² = 1/y²² → so answer is x⁸ / y²²

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Problem ⑩: (p³q⁵ / r⁷) · (p²r⁰q³ / p⁴r²)³

First, note that r⁰ = 1, so we can ignore it.

Simplify second term first: (p²·1·q³ / p⁴r²)³ = (p²q³ / p⁴r²)³

Inside: p²/p⁴ = p^(2-4) = p⁻²
So: p⁻²q³ / r²

Raise to power 3:
(p⁻²)³(q³)³ / (r²)³ = p⁻⁶q⁹ / r⁶

Now multiply with first term:
(p³q⁵ / r) · (p⁻q⁹ / r⁶)

Multiply numerators: p³·p⁻⁶ = p³; q⁵·q⁹ = q¹⁴
Multiply denominators: r⁷·r⁶ = r¹³

So: p⁻³q¹⁴ / r¹³

Make p⁻³ positive: 1/p³ → final answer: q¹⁴ / (p³r¹³)

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Final Answer:
① a⁸
② p²⁰
③ p²
④ z⁶
⑤ q⁴
⑥ l
⑦ x¹⁴b¹⁶
⑧ a¹²b¹⁰ / c¹²
⑨ x⁸ / y²²
⑩ q¹⁴ / (p³r¹³)