#"g/cm"^3# to #"lb/L"#
Let's do the conversion one step at a time using familiar conversion factors.
We know that 1 lb = 453.6 g.
∴ #(2.51 color(red)(cancel(color(black)("g"))))/("1 cm"^3) × "1 lb"/(453.6 color(red)(cancel(color(black)("g")))) = 5.534 × 10^"-3" color(white)(l)"lb/cm"^3#
We know that #"1000 cm"^3 = "1 L"#.
∴ #(5.534 × 10^"-3"color(white)(l) "lb")/(1 color(red)(cancel(color(black)("cm"^3)))) × (1000 color(red)(cancel(color(black)("cm"^3))))/"1 L" = "5.53 lb/L"#
#"g/cm"^3# to #"kg/m"^3#
We know that 1 kg = 1000 g.
∴ #(2.51 color(red)(cancel(color(black)("g"))))/("1 cm"^3) × "1 kg"/(1000 color(red)(cancel(color(black)("g")))) = 2.51 × 10^"-3"color(white)(l) "kg/cm"^3#
We know that 1 m = 100 cm.
∴ #("1 m")^3 = ("100 cm")^3#
or
#"1 m"^3 = 10^6color(white)(l) "cm"^3#
∴ #(2.51 × 10^"-3"color(white)(l) "kg")/(1 color(red)(cancel(color(black)("cm"^3)))) × (10^6 color(red)(cancel(color(black)("cm"^3))))/("1 m"^3) = 2.51 × 10^3 color(white)(l)"kg/m"^3#
