Quotient Rule of Exponents Worksheets (printable, online, answers ... - Free Printable

You're working on a Laws of Exponents: Quotient Rule worksheet.

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✔ Quotient Rule for Exponents:

> When dividing two powers with the same base, you subtract the exponents:
>
> \[
> \frac{a^m}{a^n} = a^{m - n}
> \]
>
> (This applies whether written as division `÷` or as a fraction.)

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Let’s solve each problem step by step:

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🔹 Left Column:

#### 1. \( x^6 \div x^4 \)
→ Same base \(x\), subtract exponents:
\( x^{6 - 4} = x^2 \)

✔ Answer: \( x^2 \)

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#### 2. \( y^{12} \div y^3 \)
→ Subtract exponents:
\( y^{12 - 3} = y^9 \)

✔ Answer: \( y^9 \)

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#### 3. \( \frac{x^6}{x^4} \)
→ Same as above:
\( x^{6 - 4} = x^2 \)

✔ Answer: \( x^2 \)

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#### 4. \( \frac{y^5}{y} \)
→ Note: \( y = y^1 \)
So, \( y^{5 - 1} = y^4 \)

✔ Answer: \( y^4 \)

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#### 5. \( \frac{w^2}{w^5} \)
→ Subtract exponents: \( w^{2 - 5} = w^{-3} \)
But usually we write with positive exponents:
\( w^{-3} = \frac{1}{w^3} \)

✔ Answer: \( \frac{1}{w^3} \)

*(Note: Some teachers accept negative exponents; if not specified, simplify to positive exponent form.)*

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🔹 Right Column:

#### 6. \( x^9 \div x^4 \)
→ \( x^{9 - 4} = x^5 \)

✔ Answer: \( x^5 \)

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#### 7. \( y^3 \div y \)
→ \( y^{3 - 1} = y^2 \)

✔ Answer: \( y^2 \)

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#### 8. \( \frac{m^9}{m^2} \)
→ \( m^{9 - 2} = m^7 \)

✔ Answer: \( m^7 \)

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#### 9. \( \frac{x^6}{x} \)
→ \( x^{6 - 1} = x^5 \)

✔ Answer: \( x^5 \)

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#### 10. \( \frac{y}{y^7} \)
→ \( y^{1 - 7} = y^{-6} = \frac{1}{y^6} \)

✔ Answer: \( \frac{1}{y^6} \)

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## 📝 Final Answers (Boxed for clarity):

| Problem | Answer |
|--------|--------|
| \( x^6 \div x^4 \) | \( \boxed{x^2} \) |
| \( y^{12} \div y^3 \) | \( \boxed{y^9} \) |
| \( \frac{x^6}{x^4} \) | \( \boxed{x^2} \) |
| \( \frac{y^5}{y} \) | \( \boxed{y^4} \) |
| \( \frac{w^2}{w^5} \) | \( \boxed{\frac{1}{w^3}} \) |
| \( x^9 \div x^4 \) | \( \boxed{x^5} \) |
| \( y^3 \div y \) | \( \boxed{y^2} \) |
| \( \frac{m^9}{m^2} \) | \( \boxed{m^7} \) |
| \( \frac{x^6}{x} \) | \( \boxed{x^5} \) |
| \( \frac{y}{y^7} \) | \( \boxed{\frac{1}{y^6}} \) |

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✔ Key Takeaway: Always check that the bases are the same — then just subtract the exponents! If the exponent becomes negative, rewrite as a fraction to make it positive (unless instructed otherwise).

Let me know if you want to practice more or need help with the product rule or power rule too!