Beat queue, you say, to disappear outside (8)
I believe the answer is:
vanquish
'beat' is the definition.
(I know that beat can be written as vanquish)
'queue you say to disappear outside' is the wordplay.
'queue' becomes 'q' (this might be a standard abbreviation which I don't know about).
'you say' becomes 'u' ('you' as it sounds).
'to disappear' becomes 'vanish' (vanishing is a kind of disappearing).
'outside' is an insertion indicator.
'q'+'u'='qu'
'qu' inserted into 'vanish' is 'VANQUISH'.
(Other definitions for vanquish that I've seen before include "Best" , "Defeat, conquer" , "Defeat, overcome someone" , "Come out better in a contest" , "Defeat in battle" .)