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## Homework Statement

Let F be a field. For any a,b [tex]\in[/tex] F, b[tex]\neq[/tex]0, we write a/b for ab^-1. Prove the following statements for any a,a' [tex]\in[/tex]F and b,b' [tex]\in[/tex] F\{0}:

i.) a/b = a'/b' if and only if ab' = a'b.

ii.) a/b + a'/b' = (ab' + a'b)/bb'

iii.) (a/b)(a'/b') = (aa')/(bb')

iv.) ((a/b)/(a'/b')) = (ab')/(a'b) if in addition a' [tex]\neq[/tex] 0

## Homework Equations

## The Attempt at a Solution

Okay im getting really confused in these questions because i dont know when it means to divide by and when it is showing that a/b = ab^-1

i.) want to show a/b = a'/b' if and only if ab' = a'b.

from ab' = a'b a = a'b/b' and a' = ab'/b

So substituting in values of a and a'

a/b = (a'b/b')/(ab'/b)

Carrying out division:

(a'ba')/(b'ab') and because a'b = b'a, substitute this in and cancel like terms:

(ab'a'/ab'b') = a'/b'

therefore a'/b' = a/b

is this even right because its just rearranging stuff?

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