5.6 Rules of Exponents and Scientific Notation 271 Some calculators will require you to enter the negative sign using a + −/ key after entering the exponent. Some calculators will display the answer in scientific notation: 7.77E4, which means × 7.77 104 and equals 77,700. MATHEMATICS TODAY What’s the Difference Between Debt and Deficit? We often hear economists and politicians talk about such things as revenue, expenditures, deficit, surplus, and national debt. Revenue is the money a government collects annually, mostly through taxes. Expenditures are the money a government spends annually. If revenue exceeds expenditures, a surplus occurs; if expenditures exceed revenue, a deficit occurs for that year. The national debt is the total of all the budget deficits and (the few) surpluses encountered by the U.S. federal government for more than 200 years. In 2023, the national debt was about $32.11 trillion, or about × $3.211 10 . 13 This amount is owed to investors worldwide— perhaps including you—who own U.S. government bonds. Why This Is Important Understanding the financial structure of our government is an important part of understanding our democratic system. The same principles discussed here also apply to our own finances. m A World War I–era poster encouraging investment in U.S. government bonds Example 11 Use Scientific Notation on a Calculator to Determine a Quotient Divide 0.000000000048 24,000,000,000 using a scientific calculator. Write the answer in scientific notation. Solution We first rewrite the numerator and denominator using scientific notation. See Example 9. = × × − 0.000000000048 24,000,000,000 4.8 10 2.4 10 11 10 Next we use a scientific calculator to perform the computation. The keystrokes are as follows: − ÷ = 4.8 EXP () 11 2.4 EXP 10 The display on the calculator is − 2. , 21 which means × − 2.0 10 . 21 7 Now try Exercise 45 Example 12 U.S. Debt per Person In 2023, the U.S. Department of the Treasury estimated the U.S. national debt to be about $32.11 trillion. In 2023, the population of the U. S. was estimated to be about 337 million. Determine the average debt, per person, by dividing the U. S. national debt by the U. S. population. Solution We first write the numbers involved using decimal notation. We then convert them to scientific notation. = = × 32.11 trillion 32,110,000,000,000 3.211 1013 337 million 337,000,000 3.37 108 = = × Next we divide × 3.211 1013 by × 3.37 10 ,8 using a scientific calculator. The keystrokes are ÷ = 3.211 EXP 13 3.37 EXP 8 The display shows 95,281.89911. This number indicates that in 2023, the U.S. government owed about $95,281.90 per person living in the United States. 7 Now try Exercise 61 Learning Catalytics Keyword: Angel-SOM-5.6 (See Preface for additional details.) Library of Congress
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