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Mastering Integer Exponents & Scientific Notation

Learn product, quotient, and power rules for exponents. Understand the role of 0 and negative exponents. Simplify expressions and use scientific notation effectively. Classroom examples included.

stephencook
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Mastering Integer Exponents & Scientific Notation

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  1. Chapter 5 Section 1

  2. Integer Exponents and Scientific Notation Use the product rule for exponents. Define 0 and negative exponents. Use the quotient rule for exponents. Use the power rules for exponents. Simplify exponential expressions. Use the rules for exponents with scientific notation. 5.1

  3. We use exponents to write products of repeated factors. For example, 25 is defined as 2 • 2 • 2 • 2 • 2 = 32. The number 5, the exponent, shows that the base 2 appears as a factor five times. The quantity 25 is called an exponential or a power. We read 25 as “2 to the fifth power” or “2 to the fifth.” Integer Exponents and Scientific Notation Slide 5.1- 3

  4. Use the product rule for exponents. Objective 1 Slide 5.1- 4

  5. Be careful not to multiply the bases. Keep the same base and add the exponents. Use the product rule for exponents. Slide 5.1- 5

  6. CLASSROOM EXAMPLE 1 Using the Product Rule for Exponents Solution: m8 • m6 = m8+6 = m14 Apply the product rule, if possible, in each case. m5 • p4 Cannot be simplified further because the bases m and p are not the same. The product rule does not apply. = (–5)(–9)(p4p5) = 45p4+5 = 45p9 (–5p4) (–9p5) (–3x2y3) (7xy4) = –21x3y7 = –21x2+1y3+4 = (–3)(7) x2xy3y4 Slide 5.1- 6

  7. Define 0 and negative exponents. Objective 2 Slide 5.1- 7

  8. The expression 00 is undefined. Define 0 and negative exponents. Slide 5.1- 8

  9. CLASSROOM EXAMPLE 2 Using 0 as an Exponent Solution: Evaluate. = 1 290 = 1 (–29)0 = – (290) –290 = –1 80 – 150 = 1 – 1 = 0 Slide 5.1- 9

  10. A negative exponent does not indicate a negative number; negative exponents lead to reciprocals. Define 0 and negative exponents. Slide 5.1- 10

  11. CLASSROOM EXAMPLE 3 Using Negative Exponents Solution: Write with only positive exponents. 6-5 (2x)-4 , x≠ 0 –7p-4, p≠ 0 Evaluate 4-1 – 2-1. Slide 5.1- 11

  12. CLASSROOM EXAMPLE 4 Using Negative Exponents Solution: Evaluate. Slide 5.1- 12

  13. Define 0 and negative exponents. Slide 5.1- 13

  14. Use the quotient rule for exponents. Objective 3 Slide 5.1- 14

  15. Be careful when working with quotients that involve negative exponents in the denominator. Write the numerator exponent, then a subtraction symbol, and then the denominator exponent. Use parentheses. Use the quotient rule for exponents. Slide 5.1- 15

  16. CLASSROOM EXAMPLE 5 Using the Quotient Rule for Exponents Solution: Apply the quotient rule, if possible, and write each result with only positive exponents. Cannot be simplified because the bases x and y are different. The quotient rule does not apply. Slide 5.1- 16

  17. Use the power rules for exponents. Objective 4 Slide 5.1- 17

  18. Use the power rules for exponents. Slide 5.1- 18

  19. CLASSROOM EXAMPLE 6 Using the Power Rules for Exponents Solution: Simplify, using the power rules. Slide 5.1- 19

  20. Use the power rules for exponents. Slide 5.1- 20

  21. CLASSROOM EXAMPLE 7 Using Negative Exponents with Fractions Solution: Write with only positive exponents and then evaluate. Slide 5.1- 21

  22. Use the power rules for exponents. Slide 5.1- 22

  23. Use the power rules for exponents. Slide 5.1- 23

  24. Simplify exponential expressions. Objective 5 Slide 5.1- 24

  25. CLASSROOM EXAMPLE 8 Using the Definitions and Rules for Exponents Solution: Simplify. Assume that all variables represent nonzero real numbers. Slide 5.1- 25

  26. CLASSROOM EXAMPLE 8 Using the Definitions and Rules for Exponents (cont’d) Solution: Simplify. Assume that all variables represent nonzero real numbers. Combination of rules Slide 5.1- 26

  27. Use the rules for exponents with scientific notation. Objective 6 Slide 5.1- 27

  28. In scientific notation, a number is written with the decimal point after the first nonzero digit and multiplied by a power of 10. This is often a simpler way to express very large or very small numbers. Use the rules for exponents with scientific notation. Slide 5.1- 28

  29. Use the rules for exponents with scientific notation. Slide 5.1- 29

  30. Use the rules for exponents with scientific notation. Converting to Scientific Notation Step 1Position the decimal point.Place a caret, ^, to the right of the first nonzero digit, where the decimal point will be placed. Step 2Determine the numeral for the exponent. Count the number of digits from the decimal point to the caret. This number gives the absolute value of the exponent on 10. Step 3Determine the sign for the exponent. Decide whether multiplying by 10n should make the result of Step 1 greater or less. The exponent should be positive to make the result greater; it should be negative to make the result less. Slide 5.1- 30

  31. CLASSROOM EXAMPLE 9 Writing Numbers in Scientific Notation 29,800,000 ^ Write the number in scientific notation. Solution: Step 1Place a caret to the right of the 2 (the first nonzero digit) to mark the new location of the decimal point. Step 2Count from the decimal point 7 places, which is understood to be after the caret. . Decimal point moves 7 places to the left 29,800,000 = 2.9,800,000 Step 3Since 2.98 is to be made greater, the exponent on 10 is positive. 29,800,000 = 2.98 × 107 Slide 5.1- 31

  32. CLASSROOM EXAMPLE 9 Writing Numbers in Scientific Notation (cont’d) 0.0000000503 ^ Write the number in scientific notation. Solution: Step 1Place a caret to the right of the 5 (the first nonzero digit) to mark the new location of the decimal point. Step 2Count from the decimal point 8 places, which is understood to be after the caret. Decimal point moves 7 places to the left 0.0000000503 = 0.00000005.03 Step 3Since 5.03 is to be made less, the exponent 10 is negative. 0.0000000503 = 5.03 × 10−8 Slide 5.1- 32

  33. Use the rules for exponents with scientific notation. Slide 5.1- 33

  34. When converting from scientific notation to standard notation, use the exponent to determine the number of places and the direction in which to move the decimal point. CLASSROOM EXAMPLE 10 Converting from Scientific Notation to Standard Notation Solution: Write each number in standard notation. 2.51 ×103 = 2510 Move the decimal 3 places to the right. –6.8 ×10−4 = –0.000068 Move the decimal 4 places to the left. Slide 5.1- 34

  35. CLASSROOM EXAMPLE 11 Using Scientific Notation in Computation Solution: Evaluate Slide 5.1- 35

  36. CLASSROOM EXAMPLE 12 Using Scientific Notation to Solve Problems Solution: The distance to the sun is 9.3 × 107mi. How long would it take a rocket traveling at 3.2 × 103 mph to reach the sun? d = rt, so It would take approximately 2.9 ×104 hours. Slide 5.1- 36

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