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Mastering the Laws of Exponents: Understanding Multiplication & Division of Powers

This resource provides a comprehensive overview of the laws of exponents, particularly focusing on the multiplication and division of powers with the same base. It explains how to group factors and add or subtract exponents when multiplying or dividing powers. Key concepts like raising powers to powers and practical applications in scientific notation are also explored, including real-world examples involving sound travel and biological measurements. This guide is essential for students looking to grasp the fundamentals of exponents.

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Mastering the Laws of Exponents: Understanding Multiplication & Division of Powers

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  1. Notes 57 Laws of Exponents

  2. The factors of a power, such as 74, can be grouped in different ways. Notice the relationship of the exponents in each product. 7 • 7 • 7 • 7 = 74 (7 • 7 • 7) • 7 = 73 • 71 = 74 (7 • 7) • (7 • 7) = 72 • 72 = 74

  3. A. 66 •63 6 6+ 3 6 9 n 5 + 7 n 12 Additional Example 1: Multiplying Powers with the Same Base Multiply. Write the product as one power. Add exponents. B. n5 •n7 Add exponents.

  4. A. 66 •63 6 6+ 3 6 9 n 5 + 7 n 12 Additional Example 1: Multiplying Powers with the Same Base Multiply. Write the product as one power. Add exponents. B. n5 •n7 Add exponents.

  5. A. 42 •44 Check It Out: Example 1 Multiply. Write the product as one power. B. x4 •x2

  6. Check It Out: Example 1 Continued Multiply. Write the product as one power. C. 15• 152 D. p2• p2

  7. 5555 5 5555 5 5 5 = = = 5 5 = 52 555 555 53 Notice what occurs when you divide powers with the same base.

  8. 5 7 3 7 5 – 3 7 2 7 10 x 9 x 10 – 9 x Think: x = x 1 Additional Example 2: Dividing Powers with the Same Base Divide. Write the quotient as one power. A. Subtract exponents. B. Subtract exponents. x

  9. Check It Out: Example 2 Divide. Write the quotient as one power. 9 9 A. 9 2 n 8 B. n 5

  10. Additional Example 3: Raising a Power to a Power Simplify. A. (54)2 (54)2 54 • 2 Multiply exponents. 58 B. (67)9 (67)9 Multiply exponents. 67 • 9 663

  11. 12 –3 2 3 12 • –3 2 3 Additional Example 3: Raising a Power to a Power Simplify. D. (172)–20 C. (172)–20 Multiply exponents. Multiply exponents. 172 • –20 17–40

  12. Check It Out: Example 3 Simplify. A. (72)–1 B. (7–1)2

  13. 4 –2 1 5 Check It Out: Example 3 Continued Simplify. D. (5–2)–3 C.

  14. Additional Example 4: Application The speed of sound at sea level is 3.4029 x 102 meters per second. A ship that is 5 kilometers offshore sounds its horn. About how many seconds will pass before a person standing on shore will hear the sound? Write your answer in scientific notation.

  15. Additional Example 4 Continued distance = rate x time 5 km = (3.4029 x 102) x t Write 5 km as meters. 5000 = (3.4029 x 102) x t 5 x 103 = (3.4029 x 102) x t Write 5000 in scientific notation. 3.4029 x 102 3.4029 x 102 5 103 1.469 x 101  t x = t 3.4029 x 102 102 It would take about 1.5 x 101 seconds for the sound to reach the shore.

  16. Check It Out: Example 4 The diameter of a red blood cell is about 7.6 × 10-4 millimeters. Thai has a slide with a 2-cm drop of red blood cells on it. Approximately how many cells are on the slide? Write your answer in scientific notation. (Hint: Find the ratio of the size of the drop, in millimeters, to the size of one cell.)

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