Let's solve each problem using the
Quotient Rule of Exponents.
🔷 Quotient Rule:
When dividing two expressions with the
same base, subtract the exponent in the denominator from the exponent in the numerator:
$$
\frac{a^m}{a^n} = a^{m - n}
$$
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We’ll apply this rule to each expression.
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Left Column:
1. $ x^6 \div x^4 $
$$
x^{6-4} = x^2
$$
✔ Answer: $ x^2 $
---
2. $ y^{12} \div y^3 $
$$
y^{12-3} = y^9
$$
✔ Answer: $ y^9 $
---
3. $ \frac{x^6}{x^4} $
$$
x^{6-4} = x^2
$$
✔ Answer: $ x^2 $
---
4. $ \frac{y^5}{y} $
Note: $ y = y^1 $, so:
$$
y^{5-1} = y^4
$$
✔ Answer: $ y^4 $
---
5. $ \frac{w^2}{w^5} $
$$
w^{2-5} = w^{-3}
$$
We can write this as a
fraction if needed (since negative exponents mean reciprocal):
$$
w^{-3} = \frac{1}{w^3}
$$
✔ Answer: $ w^{-3} $ or $ \frac{1}{w^3} $
---
Right Column:
6. $ x^9 \div x^4 $
$$
x^{9-4} = x^5
$$
✔ Answer: $ x^5 $
---
7. $ y^3 \div y $
$ y = y^1 $, so:
$$
y^{3-1} = y^2
$$
✔ Answer: $ y^2 $
---
8. $ \frac{m^9}{m^2} $
$$
m^{9-2} = m^7
$$
✔ Answer: $ m^7 $
---
9. $ \frac{x^6}{x} $
$ x = x^1 $, so:
$$
x^{6-1} = x^5
$$
✔ Answer: $ x^5 $
---
10. $ \frac{y}{y^7} $
$ y = y^1 $, so:
$$
y^{1-7} = y^{-6} = \frac{1}{y^6}
$$
✔ Answer: $ y^{-6} $ or $ \frac{1}{y^6} $
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✔ Final Answers:
| Expression | Simplified |
|----------|------------|
| $ x^6 \div x^4 $ | $ x^2 $ |
| $ y^{12} \div y^3 $ | $ y^9 $ |
| $ \frac{x^6}{x^4} $ | $ x^2 $ |
| $ \frac{y^5}{y} $ | $ y^4 $ |
| $ \frac{w^2}{w^5} $ | $ w^{-3} $ or $ \frac{1}{w^3} $ |
| $ x^9 \div x^4 $ | $ x^5 $ |
| $ y^3 \div y $ | $ y^2 $ |
| $ \frac{m^9}{m^2} $ | $ m^7 $ |
| $ \frac{x^6}{x} $ | $ x^5 $ |
| $ \frac{y}{y^7} $ | $ y^{-6} $ or $ \frac{1}{y^6} $ |
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📝 Summary:
Use the
quotient rule: subtract exponents when dividing like bases.
If the result is negative, you can write it as a fraction (positive exponent in the denominator).
Let me know if you'd like these rewritten in positive exponent form only!
Parent Tip: Review the logic above to help your child master the concept of quotient of powers worksheet.
