# 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

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

or

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

or

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

✅ 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."

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).
$${\color[rgb]{0.8, 0.0, 0.0}\boxed{\;\;\;\mathrm{in}^3\;\;\;}}\;\left(\xrightarrow[{1\;\mathrm{in}}]{2.540\;\mathrm{cm}}\right)^3\;{\color[rgb]{0.0, 0.0, 1.0}\boxed{\;\;\;\mathrm{cm}^3\;\;\;}}$$

$$33\underline3.04\;\cancel{\mathrm{in}^3}\;\times\;\left(\dfrac{2.540\;\mathrm{cm}}{1\;\cancel{\mathrm{in}}}\right)^3\;=\;\boxed{5.46\times10^3\;\mathrm{cm}^3}$$

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)?

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/