1. Power to some number is dimensionaless like power to exponential(e) is dimensionless like in equation for radioactive decay e has power

-lambda. t so the power, (-lambda. t)  is dimensionless.

2. If product of two quantities is dimensionless then both will have Dimension formula as inverse of each others, like in above case lambda will have DF inverse to that of t so DF of lambda is T-1 (T inverse ) or 1/T

3. Quantities which are ratio of similar quantities are dimensionless, e.g. relative density is ratio of density of material and density of water so relative density will have no DF(Dimensional Formula), in the same way the refractive index of a material is ratio of two velocities (that is velocity upon velocity )so it is dimensionless and poisson's ratio is the ratio of lateral strain to longitudinal strain so it is dimensionless.  SO IF YOU KNOW SOME FORMULA INVOLVING THE QUANTITY, YOU ARE NOT REQUIRED TO MEMORIZE THE DFs.

4. If two quantities are added/substracted or equated (+ - or =) they will  have same dimensional formula.

5. DF for time, total time, time duration (delta t) is same [ T ]

6. DF for velocity, speed, average speed, average velocity, change in velocity is same to that of [L] upon [T]

Best Wishes