Introduction - Basic Mathematics for Water and Wastewater Operators - page 6

kg
day

day
min

kg
min

× =

2

m
mm

2 2

mm × = m

2

3 3

gal
min

ft
gal

ft
min

× =

How are units that include exponents calculated? When written with exponents,
such as ft 3 , a unit can be left as is or put in expanded form, (ft)(ft)(ft), depending on
other units in the calculation. The point is that it is important to ensure that square
and cubic terms are expressed uniformly (e.g., sq ft, ft 2 , cu ft, ft 3 ). For dimensional
analysis, the use of exponents is preferred.
For example, let's say that we wish to convert 1400 ft 3 volume to gallons, and
we will use 7.48 gal/ft 3 in the conversions. The question becomes do we multiply
or divide by 7.48? We can use dimensional analysis to help us answer that question.
To determine if the math setup is correct, only the dimensions are used. First, try
dividing the dimensions:

3 3

ft
gal/ft

ft
gal
ft

=

3

3

Multiply the numerator and denominator to get:

6

ft
gal

So, by dimensional analysis, we have determined that if we divide the two dimen-
sions (ft 3 and gal/ft 3 ) then the units of the answer are ft 6 /gal, not gal. It is clear that
division is not the right approach to making this conversion. What would have hap-
pened if we had multiplied the dimensions instead of dividing?

gal
ft

3 3 3

ft × gal/ft = ft ×

3

Multiply the numerator and denominator to obtain:

3

ft × gal

3

ft