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4.1 Integer Powers of Ten You will evaluate powers with positive numbers, negative numbers, or zero for exponents. Essential Question: How do you evaluate a power of ten with a negative exponent? Daily Warm-Up Exercises For use with pages 195–200 Evaluate. 1. 10 2 ANSWER 100 2. 10 4 ANSWER 10,000 Order the numbers from least to greatest. 3. –5, 2, –7, 1, 10 ANSWER –7, –5, 1, 2, 10 4. 12, 14, –19, 22, –24 ANSWER –24, –19, 12, 14, 22 Activities Secret Worlds: The Universe Within Zero and Negative Powers of Ten Textbook p. 195 Definitions 100 = 1 a0 = 1 any number to the power of zero equals 1 10-n = 1 10-n is the reciprocal of 10n 10n a-n = 1 a-n is the reciprocal of an an To Convert Decimals (Standard Form) to Powers of Ten: If the number is bigger than 1: The power will equal the number of zeros If the number is less than 1, but bigger than 0: Make the number into a fraction The power will equal the opposite (negative) of the number of zeros in the denominator If the number is 1: The power is 0 Example 1 Write Powers of 10 in Standard Form a. 10 3 = 10 10 10 = 1,000 1 1 1 1 1 b. 10 –5 = = 0.00001 10 10 10 10 10 Guided Practice for Example 1 Write the power of 10 in standard form. 1. 10 5 ANSWER 100,000 2. 10 6 ANSWER 1,000,000 3. 10 –2 ANSWER 0.01 4. 10 –7 ANSWER 0.0000001 Guided Practice for Example 1 5. SCIENCE Scientists have created a microfabric using molded plastic. Its narrowest threads are 10 –3 millimeter across. Write this number in standard form. ANSWER 0.001 Example 2 Write Numbers in Standard Form as Powers of 10 There are 9 zeros after the 1 so the a. 1,000,000,000 = 10 9 exponent is 9. There are 5 = 6 – 1 zeros after the b. 0.000001 = 10 –6 decimal point, so the exponent is – 6. Example 3 Using a Negative Exponent STROBE LIGHTS The picture of the golf ball was taken using a strobe light. Each flash of the strobe light lasts about 1 microsecond. Use a power of 10 to write this time in seconds. SOLUTION Each flash lasts 1 microsecond, or equivalently, 1 second. 1,000,000 1 1 = 6 Write 1,000,000 as 106. 1,000,000 10 Example 3 Using a Negative Exponent 1 = 10 –6 Use the fact that = 10 –n. 10 n ANSWER Each flash of the strobe light lasts about 10 –6 second. Guided Practice for Examples 2 and 3 6. Write the number 0.0001 as a power of 10. ANSWER 10 –4 7. WHAT IF? In Example 3, suppose each flash of the strobe light lasts one millisecond (or one thousandth of a second). Use a power of 10 to write this time in seconds. ANSWER 10 –3 Example 4 Multiple Choice Practice Which list of numbers is ordered from least to greatest? 10 2, 10 –3, 0.0001, 1, 10 10 –3, 0.0001, 1, 10, 10 2 0.0001, 10 –3, 1, 10, 10 2 0.0001, 1, 10, 10 2, 10 –3 SOLUTION Write each number in standard form. Then compare the numbers. 10 2 = 100, 10 –3 = 0.001, 0.0001, 1, 10 Example 4 Multiple Choice Practice The order of the numbers from least to greatest is 0.0001, 10 –3, 1, 10, and 10 2. ANSWER The correct answer is C. Guided Practice for Example 4 Order the numbers from least to greatest. 8. 100, 10 –2, 1 ANSWER 10 –2, 1, 100 9. 0.01, 10 2, 1 ANSWER 0.01, 1, 10 2 10. 10, 10 0, 10 –1, 10 2 ANSWER 10 –1, 10 0, 10, 10 2 Essential Question: How do you evaluate a power of ten with a negative exponent? Independent Practice Textbook p. 198-200 (1-15, 19, 23, 29-31, 39- 41)