5.6 Rules of Exponents and Scientific Notation 265 What is the current world population? What is the diameter of an atom? What is the diameter of a galaxy? What is the wavelength of an x-ray? What is the current national federal debt? All these questions have answers that may involve either very large numbers or very small numbers. One way to accurately represent such numbers is with scientific notation. In this section, we will study how to use scientific notation when answering questions such as those posed above. SECTION 5.6 Rules of Exponents and Scientific Notation LEARNING GOALS Upon completion of this section, you will be able to: 7 Evaluate expressions using the rules of exponents. 7 Write numbers using scientific notation. 7 Use a scientific calculator to solve problems involving scientific notation. Why This Is Important Scientific notation is used in many fields of study, including economics, physics, chemistry, and astronomy. Items such as a computer hard drive’s capacity, measured in gigabytes, can also be represented using scientific notation. In Section 5.2 we introduced exponents. Here we discuss the rules of exponents, which are very important to the study of mathematics. Rules of Exponents We begin with the product rule for exponents. Consider ⋅ = ⋅ ⋅ ⋅ ⋅ = 2 2 2 2 2 2 2 2 2 3 5 2 factors 3 factors This example illustrates the product rule for exponents. Product Rule for Exponents ⋅ = + a a a m n m n Therefore, by using the product rule, ⋅ = = + 2 2 2 2 . 2 3 2 3 5 Example 1 Using the Product Rule for Exponents Use the product rule to simplify. a) ⋅ 5 52 b) ⋅ 3 3 2 3 Solution a) ⋅ = ⋅ = = = + 5 5 5 5 5 5 125 2 1 2 1 2 3 b) ⋅ = = = + 3 3 3 3 243 2 3 2 3 5 7 Now try Exercise 7 Consider = ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ = 2 2 2 2 2 2 2 2 2 2 2 2 2 5 2 3 This example illustrates the quotient rule for exponents. Alex Mit/Shutterstock
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