(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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Verify My Proof Please

**Physics Forums | Science Articles, Homework Help, Discussion**