Quotient Rule Worksheet - Simplify expressions using the quotient rule and write answers in positive exponents.

Problem Overview:

The task involves simplifying expressions using the Quotient Rule of exponents. The Quotient Rule states:

\[
\frac{a^m}{a^n} = a^{m-n}
\]

where \( a \) is the base, and \( m \) and \( n \) are the exponents. The goal is to simplify each expression and write the answers in positive exponents.

Solution:

#### 1. \( \frac{x^5}{x^2} \)
Using the Quotient Rule:
\[
\frac{x^5}{x^2} = x^{5-2} = x^3
\]

#### 2. \( \frac{x^7}{x^4} \)
Using the Quotient Rule:
\[
\frac{x^7}{x^4} = x^{7-4} = x^3
\]

#### 3. \( \frac{x^9}{x^6} \)
Using the Quotient Rule:
\[
\frac{x^9}{x^6} = x^{9-6} = x^3
\]

#### 4. \( \frac{x^8}{x^4} \)
Using the Quotient Rule:
\[
\frac{x^8}{x^4} = x^{8-4} = x^4
\]

#### 5. \( \frac{a^{10}}{a^4} \)
Using the Quotient Rule:
\[
\frac{a^{10}}{a^4} = a^{10-4} = a^6
\]

#### 6. \( \frac{b^8}{b^2} \)
Using the Quotient Rule:
\[
\frac{b^8}{b^2} = b^{8-2} = b^6
\]

#### 7. \( \frac{b^9}{b^3} \)
Using the Quotient Rule:
\[
\frac{b^9}{b^3} = b^{9-3} = b^6
\]

#### 8. \( \frac{d^7}{d^3} \)
Using the Quotient Rule:
\[
\frac{d^7}{d^3} = d^{7-3} = d^4
\]

#### 9. \( \frac{x^{18}}{x^{12}} \)
Using the Quotient Rule:
\[
\frac{x^{18}}{x^{12}} = x^{18-12} = x^6
\]

#### 10. \( \frac{w^8}{w^4} \)
Using the Quotient Rule:
\[
\frac{w^8}{w^4} = w^{8-4} = w^4
\]

#### 11. \( \frac{x^9}{x^3} \)
Using the Quotient Rule:
\[
\frac{x^9}{x^3} = x^{9-3} = x^6
\]

#### 12. \( \frac{m^6}{m^2} \)
Using the Quotient Rule:
\[
\frac{m^6}{m^2} = m^{6-2} = m^4
\]

#### 13. \( \frac{n^7}{n^3} \)
Using the Quotient Rule:
\[
\frac{n^7}{n^3} = n^{7-3} = n^4
\]

#### 14. \( \frac{x^8}{x^2} \)
Using the Quotient Rule:
\[
\frac{x^8}{x^2} = x^{8-2} = x^6
\]

#### 15. \( \frac{y^9}{y^3} \)
Using the Quotient Rule:
\[
\frac{y^9}{y^3} = y^{9-3} = y^6
\]

#### 16. \( \frac{x^6}{x^2} \)
Using the Quotient Rule:
\[
\frac{x^6}{x^2} = x^{6-2} = x^4
\]

#### 17. \( \frac{x^8}{x^2} \)
Using the Quotient Rule:
\[
\frac{x^8}{x^2} = x^{8-2} = x^6
\]

#### 18. \( \frac{x^{10}}{x^5} \)
Using the Quotient Rule:
\[
\frac{x^{10}}{x^5} = x^{10-5} = x^5
\]

Final Answers:

\[
\boxed{
\begin{array}{lll}
1. & x^3 & 2. & x^3 & 3. & x^3 \\
4. & x^4 & 5. & a^6 & 6. & b^6 \\
7. & b^6 & 8. & d^4 & 9. & x^6 \\
10. & w^4 & 11. & x^6 & 12. & m^4 \\
13. & n^4 & 14. & x^6 & 15. & y^6 \\
16. & x^4 & 17. & x^6 & 18. & x^5 \\
\end{array}
}
\]