Let's go through each expression step-by-step and apply the appropriate exponent rules:
1. Expression: [tex]\( x \cdot x \)[/tex]
- Rule Used: Product of powers with the same base: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]
- Simplification: [tex]\( x^2 \)[/tex]
2. Expression: [tex]\( y \cdot y \cdot y \cdot y \)[/tex]
- Rule Used: Product of powers with the same base: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]
- Simplification: [tex]\( y^4 \)[/tex]
3. Expression: [tex]\( x \cdot x \cdot y \cdot y \cdot y \)[/tex]
- Rule Used: Product of powers with the same base: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex] and grouping terms with the same base.
- Simplification: [tex]\( x^2 y^3 \)[/tex]
4. Expression: [tex]\( y \cdot z \cdot z \cdot z \cdot z \cdot z \)[/tex]
- Rule Used: Product of powers with the same base: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex] and grouping terms with the same base.
- Simplification: [tex]\( y \cdot z^5 \)[/tex]
5. Expression: [tex]\( x^2 \cdot x^3 \)[/tex]
- Rule Used: Product of powers with the same base: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]
- Simplification: [tex]\( x^5 \)[/tex]
6. Expression: [tex]\( x^5 \cdot x^4 \)[/tex]
- Rule Used: Product of powers with the same base: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]
- Simplification: [tex]\( x^9 \)[/tex]
7. Expression: [tex]\( y^6 \cdot y \)[/tex]
- Rule Used: Product of powers with the same base: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]
- Simplification: [tex]\( y^7 \)[/tex]
8. Expression: [tex]\( (x^4)^3 \)[/tex]
- Rule Used: Power of a power: [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]
- Simplification: [tex]\( x^{12} \)[/tex]
9. Expression: [tex]\( (y^3)^2 \)[/tex]
- Rule Used: Power of a power: [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]
- Simplification: [tex]\( y^6 \)[/tex]
10. Expression: [tex]\( x^2 \cdot x \cdot y^3 \cdot y^4 \)[/tex]
- Rule Used: Product of powers with the same base: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex] and grouping terms with the same base.
- Simplification: [tex]\( x^3 y^7 \)[/tex]
11. Expression: [tex]\( a^4 \cdot b^8 \cdot a^5 \cdot b^2 \)[/tex]
- Rule Used: Product of powers with the same base: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex] and grouping terms with the same base.
- Simplification: [tex]\( a^9 b^{10} \)[/tex]
12. Expression: [tex]\( c^3 \cdot d \cdot c^4 \cdot b \)[/tex]
- Rule Used: Product of powers with the same base: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex] and grouping terms with the same base.
- Simplification: [tex]\( c^7 d b \)[/tex]
13. Expression: [tex]\( \frac{x^5}{x^2} \)[/tex]
- Rule Used: Quotient of powers with the same base: [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]
- Simplification: [tex]\( x^3 \)[/tex]
14. Expression: [tex]\( \frac{y^8}{y^3} \)[/tex]
- Rule Used: Quotient of powers with the same base: [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]
- Simplification: [tex]\( y^5 \)[/tex]
Finally, let's present these results in the tabular form.
[tex]\[\begin{array}{|c|c|}
\hline \text{EXPRESSION} & \text{SIMPLIFIED} \\
\hline x \cdot x & x^2 \\
\hline y \cdot y \cdot y \cdot y & y^4 \\
\hline x \cdot x \cdot y \cdot y \cdot y & x^2 y^3 \\
\hline y \cdot z \cdot z \cdot z \cdot z \cdot z & y \cdot z^5 \\
\hline x^2 \cdot x^3 & x^5 \\
\hline x^5 \cdot x^4 & x^9 \\
\hline y^6 \cdot y & y^7 \\
\hline \left(x^4\right)^3 & x^{12} \\
\hline \left(y^3\right)^2 & y^6 \\
\hline x^2 \cdot x \cdot y^3 \cdot y^4 & x^3 y^7 \\
\hline a^4 \cdot b^8 \cdot a^5 \cdot b^2 & a^9 b^{10} \\
\hline c^3 \cdot d \cdot c^4 \cdot b & c^7 d b \\
\hline \frac{x^5}{x^2} & x^3 \\
\hline \frac{y^8}{y^3} & y^5 \\
\hline
\end{array}\][/tex]
