To determine the correct pairs of equivalent expressions, let's match them one at a time.
Given the list of expressions:
1. [tex]\( \frac{b^e}{a^6} \)[/tex]
2. [tex]\( \frac{a^6}{b^6} \)[/tex]
3. [tex]\( \frac{a^6}{8^6} \)[/tex]
4. [tex]\( a^5 b^3 \)[/tex]
5. [tex]\( \frac{b^3}{a^b} \)[/tex]
6. [tex]\( \frac{b^e}{a^b} \)[/tex]
7. [tex]\( \frac{a^a}{b^a} \)[/tex]
8. [tex]\( \frac{1}{a^8 b^4} \)[/tex]
9. [tex]\( \frac{a^4 b^{-2}}{a^{-2} b^3} \)[/tex]
10. [tex]\( \frac{a^3 b^{-4}}{a^{-3} b^2} \)[/tex]
11. [tex]\( \frac{a^{-4} b^{-2}}{a^2 b^2} \)[/tex]
We need to pair them such that each pair consists of equivalent expressions. The pairs are:
1. [tex]\( \frac{a^6}{b^6} \)[/tex] is equivalent to [tex]\( \frac{b^e}{a^b} \)[/tex].
2. [tex]\( \frac{1}{a^8 b^4} \)[/tex] is equivalent to [tex]\( \frac{a^{-4} b^{-2}}{a^2 b^2} \)[/tex].
3. [tex]\( a^5 b^3 \)[/tex] is equivalent to [tex]\( \frac{a^6}{8^6} \)[/tex].
4. [tex]\( \frac{a^4 b^{-2}}{a^{-2} b^3} \)[/tex] is equivalent to [tex]\( 1_{--1}^1 \)[/tex].
5. [tex]\( \frac{a^3 b^{-4}}{a^{-3} b^2} \)[/tex] currently has no identified equivalent match from the provided list.
Now, confirm each pairing:
- [tex]\( \frac{a^6}{b^6} \)[/tex] matches [tex]\( \frac{b^e}{a^b} \)[/tex]
- [tex]\( \frac{1}{a^8 b^4} \)[/tex] matches [tex]\( \frac{a^{-4} b^{-2}}{a^2 b^2} \)[/tex]
- [tex]\( a^5 b^3 \)[/tex] matches [tex]\( \frac{a^6}{8^6} \)[/tex]
- [tex]\( \frac{a^4 b^{-2}}{a^{-2} b^3} \)[/tex] matches [tex]\( 1_{--1}^1 \)[/tex]
- One expression [tex]\( \frac{a^3 b^{-4}}{a^{-3} b^2} \)[/tex] remains unmatched.
Thus, the pairs are:
- [tex]\( \frac{a^6}{b^6} \)[/tex] and [tex]\( \frac{b^e}{a^b} \)[/tex]
- [tex]\( \frac{1}{a^8 b^4} \)[/tex] and [tex]\( \frac{a^{-4} b^{-2}}{a^2 b^2} \)[/tex]
- [tex]\( a^5 b^3 \)[/tex] and [tex]\( \frac{a^6}{8^6} \)[/tex]
- [tex]\( \frac{a^4 b^{-2}}{a^{-2} b^3} \)[/tex] and [tex]\( 1_{--1}^1 \)[/tex]
These are the correct matched pairs for equivalent expressions.
