2.8: Units Raised to a Power

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⚙️ Learning Objectives

To convert a value reported in one unit raised to a power of 10, to a corresponding value in a different unit raised to the same power of 10, using conversion factors.

Conversion factors for area and volume may also be produced using dimensional analysis. Just remember that if a quantity is raised to a power of 10, both the number and the unit must be raised to the same power of 10. For example, to convert 1500 cm² to m², we need to start with the relationship between centimeter and meter. We know that 1 cm = 10^–2 m or 100 cm = 1 m, but since we are given the quantity in 1500 cm², then we have to use the relationship:

1 cm² = (10^–2 m)² = 10^–4 m²

CONCEPT MAP

\({\color[rgb]{0.8, 0.0, 0.0}\boxed{\;\;\mathrm{cm}^2\;\;}}\;\left(\xrightarrow[{1\;\mathrm{cm}}]{10^{-2}\;\mathrm m}\right)^2\;{\color[rgb]{0.0, 0.0, 1.0}\boxed{\;\;\;\mathrm m^2\;\;\;}}\)

CALCULATION

\(1500\:\cancel{\text{cm}^2}\times\left(\dfrac{10^{-2}\:\text{m}}{1\:\cancel{\text{cm}}}\right)^2=\boxed{0.15\:\text{m}^2}\)

\(1500\:\cancel{\text{cm}^2}\times\left(\dfrac{1\:\text{m}}{100\:\cancel{\text{cm}}}\right)^2=\boxed{0.15\:\text{m}^2}\)

\(1500\:\cancel{\text{cm}^2}\times\dfrac{1\:\text{m}^2}{10,000\:\cancel{\text{cm}^2}}=\boxed{0.15\:\text{m}^2}\)

✅ Example \(\PageIndex{1}\)

What is the volume of a sphere (radius = 4.30 inches) in cubic centimeters (cm³), given \(V=\frac43\mathrm\pi r^3\)?

Solution

Steps for Problem Solving	Result
Identify the "given” information and what the problem is asking you to "find."	Given: radius = 4.30 in Find: cm³ (volume)
List other known quantities. Metric prefixes are found in 2.5: The Metric System. Other conversions may be found in Appendix 2: Conversions.	\(V=\frac43\mathrm\pi r^3\) = \(\frac43\times\;3.1416\;\times(4.3\underline0\;\mathrm{in})^3\) = \(33\underline3.04\;\mathrm{in}^3\) 1 in = 2.540 cm
Prepare a concept map and use the proper conversion(s).	\({\color[rgb]{0.8, 0.0, 0.0}\boxed{\;\;\;\mathrm{in}^3\;\;\;}}\;\left(\xrightarrow[{1\;\mathrm{in}}]{2.540\;\mathrm{cm}}\right)^3\;{\color[rgb]{0.0, 0.0, 1.0}\boxed{\;\;\;\mathrm{cm}^3\;\;\;}}\)
Calculate the answer.	\(33\underline3.04\;\cancel{\mathrm{in}^3}\;\times\;\left(\dfrac{2.540\;\mathrm{cm}}{1\;\cancel{\mathrm{in}}}\right)^3\;=\;\boxed{5.46\times10^3\;\mathrm{cm}^3}\)
Think about your result.	A centimeter is a smaller unit than an inch, so the answer in cubic centimeters is larger than the given value in cubic inches.

✏️ Exercise \(\PageIndex{1}\)

Lake Tahoe has a surface area of 191 square miles. What is the area in square kilometers (km²)?

Answer: 494 km²

This page is shared under a CK-12 license and was authored, remixed, and/or curated by Melissa Alviar-Agnew, Henry Agnew, and Lance S. Lund (Anoka-Ramsey Community College). Original source: https://www.ck12.org/c/chemistry/.

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