Question

prove that \[(ab)^{-1} = a^{-1}b^{-1}\] prove that \[(ab)^{-1} = a^{-1}b^{-1}\] @Mathematics

Answer 1

\[(ab)^{-1}(ab)=1\]
\[aa^{-1}b^{-1}b=1\]
then this means
\[(ab)^{-1}(ab)=aa^{-1}b^{-1}b\]
multiplication is commutative so we can write
\[(ab)^{-1}(ab)=a^{-1}b^{-1}ab\]
\[(ab)^{-1}(ab)=a^{-1}b^{-1}(ab)\]
since ab=ab
then this must mean that
\[(ab)^{-1}=a^{-1}b^{-1}\]

Answer 2

my book game me this one-liner:\[ab(a^{-1}b^{-1}) = (a*a^{-1})(b*b^{-1}) = 1; \ \ hence\ \ a^{-1}b^{-1} = (ab)^{-1}\]

Answer 3

looks similar to mine

Answer 4

but he started a little different.

Answer 5

they did the whole equals to one thing

Answer 6

\[1 = ab*(ab)^{-1}\]