(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that [tex](ab)^{-1} = a^{-1}b^{-1}[/tex], if, [tex]a,b \neq 0[/tex]

2. Relevant properties

Associative property of multiplication

Existence of multiplicative inverses

3. The attempt at a solution

Since [tex]a,b \neq 0[/tex], there exists a number [tex](ab)^{-1}[/tex] such that [tex](ab)^{-1}\cdot(ab) = 1[/tex]. Multiplying both sides by [tex]a^{-1}b^{-1}[/tex], we have

[tex](ab)^{-1}\cdot(ab) \cdot (a^{-1}b^{-1}) = 1\cdot (a^{-1}b^{-1}) \Rightarrow (ab)^{-1}\cdot(a \cdot a^{-1})\cdot(b \cdot b^{-1}) = a^{-1}b^{-1} \Rightarrow (ab)^{-1}\cdot 1 \cdot 1 = a^{-1}b^{-1} \Rightarrow (ab)^{-1} = a^{-1}b^{-1}[/tex]

Any comments on making this shorter or neater or correct would be much appreciated.

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# Homework Help: Verify My Proof Please

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