“I must surely defend myself and attempt to uproot from your minds in so short a time the slander that has resided there so long.”

The quote is saying that the court proceedings may be only for a short period of time, the amount of time that the information that is being discussed has taken place for many years. The amount of time that Socrates accusers have been collecting   information and possible evidence to support the accusations have been an ongoing thing for many years. Socrates tries from the very beginning to breakdown a lifetime of preconceived notions and morals that have allowed the jury, along with others witnessing this trial, to come to the final decision without considering anything else.
Socrates believes that in order to defend himself properly, he must try to change the minds of those who had heard him when they were at a young age and decided that what they heard wasn’t truthful based upon what they had been taught by others up until then. Socrates also has to dispel the rumors that have come out of what others had heard and twisted when repeating it to others throughout the days and years. The accusers that are being believed above others are children, since they are considered pure of heart and have not been corrupted by life. No matter how much time has past, long or short, since they had heard Socrates, there will never be enough time to defend what others believe in against your own beliefs.
I believe that this quote states that a thought can take root in someone within a moments notice and stay there forever. When you try to break that thought process and belief apart, the amount of time that you may have is never going to be long enough. Children are the most susceptible to believing something that may or may not be correct as soon as they see or hear it. Now try later in time to change what they believe is right and you will never have enough time to make that change. You can compare this quote to the way long and short term memory...