Chapter 2 · Measurement and Problem Solving
2.9Units Raised to a Power
12 min · two checks
Predict
There are 100 cm in 1 m. How many cubic centimeters are in one cubic meter?
The idea
- Cube or square a conversion factor correctly.
- Avoid the classic “100 cm³ = 1 m³” error.
When a unit is squared or cubed, the conversion factor is squared or cubed with it. 1 m = 100 cm, so 1 m² = (100 cm)² = 10,000 cm², and 1 m³ = (100 cm)³ = 1,000,000 cm³. The exponent applies to the number and to the unit. Forgetting it undercounts volumes by thousands or millions, which is exactly how dosage and concrete-volume errors get made.
A cubic centimeter is a milliliter, so a cubic meter is 1000 L, because 1,000,000 mL = 1000 L. In a chain, raise only the length factor that is being converted into an area or a volume. Significant figures still come from the measured quantity; the 100 in 100 cm = 1 m is exact.
Keep these
- Square the factor for areas; cube it for volumes.
- 1 m³ = 10⁶ cm³ = 1000 L.
- 1 cm³ = 1 mL exactly.
Worked path
A dose is prepared from a solution, and a student needs the volume 0.250 m³ expressed in liters.
Observe
0.250 m³ has three significant figures. Want liters.