The density of liquid mercury is 13.6g/ml. What is it's density in units of lb/in³ ?

Use a couple of conversion factors to help you with that.

is defined as mass per unit of volume.

In your case, the density of mercury is given in grams per mililiters. In order to get the units to pounds per cubic inch, you will need to use a couple of conversion factors.

First, use the fact that 1 pound is equivalent to 453.59237 grams. To get from mililiters to cubic inches, you will use the fact that ##"1 mL" = "1 cm"""^3##, and that ##"1 in" = "2.54 cm"##.

To determine how much one cubic inch is, you can use

##"1 in"""^3 = "1 in" xx "1 in" xx "1 in"##

Now replace the inch with its equivalent in centimeters to get

##"1 in"""^3 = "2.54 cm" xx "2.54 cm" xx "2.54 cm" = "16.387064 cm"""^3##

So, your conversion will look like this

##13.6(color(red)(cancel(color(black)("g"))))/(1color(red)(cancel(color(black)("mL")))) * (1color(red)(cancel(color(black)("mL"))))/(1color(red)(cancel(color(black)("cm"^3)))) * "1 lb"/(453.59237color(red)(cancel(color(black)("g")))) * (16.387064color(red)(cancel(color(black)("cm"^3))))/("1 in"^3) = color(green)(0.491"lb"/"in"^3)##

The answer is rounded to three , the number of sig figs you have for the density in grams per mililiter.