6 Squared and Cubed Units
Some units are squared, such as an area of 150 m2, or cubed, such as a volume of 25 cm3. Remember that m2 means m*m, and cm3 means cm*cm*cm. When performing unit conversions on a unit raised to a power, you need to remember to raise the conversion factor to the power as well. That is, you need to convert both meters in m2, or all three centimeters in cm3. This is illustrated in the following examples.
Example 9.1
Convert 23.0 m/s2 to m/min2.
Answer: Since the units are a fraction, write the given quantity as a fraction
[latex]\frac{23.0m}{s^2}[/latex]
You can work with the units independently, so you will leave m alone. The conversion needed is 60 s = 1 min. The conversion can be squared (remembering to square the 60 when entering into the calculator)
[latex]\frac{23.0m}{s^2}*(\frac{60s}{1min})^2=82800m/min^2[/latex]
or the conversion could be used in one dimension two times
[latex]\frac{23.0m}{s^2}*\frac{60s}{1min}*\frac{60s}{1min}=82800m/min^2[/latex]
Example 9.2
Convert 4.5 m3 to cm3.
Answer: Start with the given quantity and use the needed conversion (100 cm = 1 m) for each meter in m3. The conversion can be cubed (remembering to cube the 100 when entering into the calculator)
[latex]4.5m^3*(\frac{100cm}{1m})^3=4.5x10^6cm^3[/latex]
or the conversion can be used in one dimension three times
[latex]4.5m^3*\frac{100cm}{1m}*\frac{100cm}{1m}*\frac{100cm}{1m}=4.5x10^6cm^3[/latex]
In chemistry we will use cm3 for volume or the volume unit L (with or without metric prefixes). When converting between these units, you should know that 1 cm3 = 1 mL. Note that the centimeter is cubed but the milliliter is not. Therefore, in the following example going from m3 to cm3 requires a cubed conversion factor but going from mL to L does not.
Example 9.3
Convert 73 m3 to L.
Answer: Start with the given quantity. The needed conversions are 1 m = 100 cm, 1 cm3 = 1 mL, and 1000 mL = 1 L.
[latex]73m^3*(\frac{100cm}{1m})^3*\frac{1mL}{1cm^3}*\frac{1L}{1000mL}=7.3x10^4L[/latex]
