# Group Product/Examples/b a^-1 x a b^-1 = b a

## Examples of Operations on Product Elements

Solve for $x$ in:

$b a^{-1} x a b^{-1} = b a$

## Solution

 $\ds b a^{-1} x a b^{-1}$ $=$ $\ds b a$ $\ds \leadsto \ \$ $\ds b^{-1} b a^{-1} x a b^{-1}$ $=$ $\ds b^{-1} b a$ Product of both sides with $b^{-1}$ $\ds \leadsto \ \$ $\ds b^{-1} b a^{-1} x a b^{-1} b$ $=$ $\ds b^{-1} b a b$ Product of both sides with $b$ $\ds \leadsto \ \$ $\ds a^{-1} x a$ $=$ $\ds a b$ Group Axiom $\text G 3$: Existence of Inverse Element $\ds \leadsto \ \$ $\ds a a^{-1} x a$ $=$ $\ds a^2 b$ Product of both sides with $a$ $\ds \leadsto \ \$ $\ds a a^{-1} x a a^{-1}$ $=$ $\ds a^2 b a^{-1}$ Product of both sides with $a^{-1}$ $\ds \leadsto \ \$ $\ds x$ $=$ $\ds a^2 b a^{-1}$ Group Axiom $\text G 3$: Existence of Inverse Element

$\blacksquare$