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Math 9: Laws of Exponents

Math 9: Laws of Exponents. Products of Exponents Quotients of Exponents Negative Exponents Evaluating Exponents Scientific Notation. 1. Products of Exponents. We know that 5 3 = 5∙5∙5 So, 5 3 ∙ 5 4 = 5 ∙5∙5 ∙ 5∙5∙5∙5 5 3 ∙ 5 4 = 5 7

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Math 9: Laws of Exponents

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  1. Math 9: Laws of Exponents Products of Exponents Quotients of Exponents Negative Exponents Evaluating Exponents Scientific Notation

  2. 1. Products of Exponents • We know that 53= 5∙5∙5 • So, 53∙ 54 = 5∙5∙5 ∙ 5∙5∙5∙5 • 53∙ 54 = 57 • Do you see a relationship between 53∙ 54 and 57?

  3. Exponent Law #1 For any real numbers a, m, and n:

  4. Practice Problems Now do practice problems 1 through 4 on your handout. We will go over them when you are done.

  5. Now, go a step further… what happens if we have ? That would mean: From Exponent Law #1 we know that,

  6. Do you see a relationship between and ?

  7. Exponent Law #2 For any real numbers a, m, and n:

  8. When there is a product raised to a power, raise each factor to the outside power. Example:

  9. Exponent Law #3 For any real numbers a, b, k, and m:

  10. Practice Problems • Now do practice problems 5 through 8 on your handout. • We will go over them when you are done.

  11. Math 9: Laws of Exponents Products of Exponents Quotients of Exponents Negative Exponents Evaluating Exponents Scientific Notation

  12. 2. Dividing with exponents Do you see a relationship between and ?

  13. Exponent Law #4 For any real numbers a, k, and m, where a ≠ 0: Note: the same base number in the numerator and denominator

  14. Exponent Law #4 Corollary Any real number raised to the zero power is 1. and by Law #4 So,

  15. Exponent Law #5 For any real numbers a, b, k, m and w: and

  16. Practice Problems • Now do practice problems 9 through 11 on your handout. • We will go over them when you are done.

  17. Math 9: Laws of Exponents Products of Exponents Quotients of Exponents Negative Exponents Evaluating Exponents Scientific Notation

  18. 3. Negative Exponents Consider and use Law #4. But what does mean?

  19. Thus, Which leads us to the next exponent law…

  20. Exponent Law #6 For any real numbers x, m, and k, where x ≠ 0: and However, this law comes with a warning….

  21. CAUTION!! WARNING!! It is very easy to make mistakes! What is wrong with… ?

  22. CAUTION!! WARNING!! Be careful with the minus sign… Not

  23. CAUTION!! WARNING!! Another easy sign error… What’s wrong with…. ?

  24. CAUTION!! WARNING!! Be careful with the negative signs… Not

  25. Practice Problems • Now do practice problems 12 – 14 on your handout. • We will go over them when you are finished.

  26. More Practice Problems • Now you can do practice problems 15 through 18 on your handout. • These problems will use all the Exponent Laws we have learned so far. • We will go over them when you are done.

  27. Math 9: Laws of Exponents Products of Exponents Quotients of Exponents Negative Exponents Evaluating Exponents Scientific Notation

  28. Consider this equation: Reduce the fraction: Divide both sides by 2: Now, what times itself 3 times equals 729? Thus, x = 9

  29. Practice Problems • Now do practice problems 19 through 21 on your handout. • We will go over them when you are done.

  30. Math 9: Laws of Exponents Products of Exponents Quotients of Exponents Negative Exponents Evaluating Exponents Scientific Notation

  31. 5. Scientific Notation • Used for very large & very small numbers. • Makes multiplying & dividing much easier. • Has this form: • Has 1 non-zero digit left of the decimal point. • Move the decimal point counting the moves. • Moving left means a positive exponent. • Moving right means a negative exponent.

  32. Write 8,532,000 in scientific notation. • Locate the decimal point. • Move the decimal point. • Count number of places moved and direction. • Make the number of places moved the exponent. • Write the number.

  33. Write 0.0000345 in scientific notation. • Locate the decimal point. • Move the decimal point. • Count the number of places moved and direction. • Make the number of places moved the exponent. • Write the number.

  34. Multiplying and Dividing with scientific notation. • Consider : • Separate the decimals and powers of 10. • Multiply decimals and powers of 10 separately.

  35. Another example: • Consider • Put numbers into scientific notation. • Separate decimals and powers of 10. Then multiply. WARNING!! THIS IS NOT SCIENTIFIC NOTATION!

  36. Put 26.46 x into scientific notation. • Move the decimal counting places moved and direction. • Thus,

  37. Practice Problems • Now do practice problems 22 through 27 on your handout. • We will go over them when you are done.

  38. Review Lessons Worksheet • You are now ready to do the Worksheet for these lessons. • It is worth 20 points toward passing Math 9. • When is it due???

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