What is dimensional units/quantity and dimensional state First, I am not a native English-speaking student so I am not good at physics definitions in English. I participated in the MIT e-learning course on classical physics. The 1st lesson is about 3 fundamental physical quantities (time, length and mass).

It mentions dimensional units/quantity and dimensional state. I couldn't find the meaning of those in a dictionary. Can someone give me an specific explanation?
Also, in the lesson, why are the quantities always associated with some power of unknown value like $\alpha,\beta$, etc when we predict the equation? And how do we know what quantities should be added into the equation of a model?

 A: I'm not sure exactly what you are asking, so I will simply write some relevant facts that might answer your question.
Dimensional analysis is a powerful tool for solving problems in physics. If we want the formula for a quantity $Q$, we can guess the formula for $Q$ by first writing a product of all relevant dimensional quantities raised to unknown powers $\alpha,\beta,\gamma\ldots$ and second making sure that our formula has the same dimension on the left and right-hand sides and solve for $\alpha,\beta,\gamma\ldots$
It might be the case that there exists a dimensionless combination of the relevant dimensional quantities. In that case, our method is not as useful. 
A: Perhaps you are confused between dimension and unit.
Note that $cm$ and $m$ are different units but have same dimension of length. See? It's simple. They have only different magnitudes.
You have to understand that you cannot subtract or add 1 kg from 1 metre. Makes no sense, right?
Suppose you want to know about speed. You know that it is $\frac {distance}{time}$ Hence its units are  $\frac {m}{s}$ and its dimensions are clear by formula.
You see that if a formula says that $1 kg = 1 s$ , It makes no sense, right?
So you check what the thing you wanna find about depends on and let analyse how to multiply and divide them to get the same dimension of thing you are looking for.
Note that this still will not give you perfect formula as $\frac {distance}{time}$ and $2*\frac {distance}{time}$ have same dimensions, you will be short of a constant.
Also it cannot predict equations like $v=u+at$
