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Mastering Exponents and Scientific Notation: A Comprehensive Guide

This guide covers the fundamentals of exponents and scientific notation, focusing on multiplying and dividing powers with like bases, negative exponents, and the exponent zero. Through practical examples and practice exercises, learners will gain clarity on simplifying expressions using exponents, raising a power to a power, and multiplying and dividing monomials. Each section includes step-by-step examples followed by practice problems to reinforce understanding and mastery of these key mathematical concepts.

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Mastering Exponents and Scientific Notation: A Comprehensive Guide

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  1. B11Exponents and Scientific Notation

  2. Multiplying Powers with Like Bases For any rational number a, and for all whole numbers m and n,

  3. Example 1 Simplify. Express using exponents. a) b) c) d)

  4. Practice Simplify. Express using exponents. a) b) c) d)

  5. Dividing Powers with Like Bases For any rational number a except 0, and for all whole numbers m and n,

  6. Example 2 Simplify. Express using exponents. a) b) c)

  7. Practice Simplify. Express using exponents. a) b) c)

  8. Negative Exponents For any rational number a except 0, and for all whole numbers m and n,

  9. Example 3 Express using positive exponents. a) b) c) d)

  10. Practice Express using exponents. 1) 2) 3)

  11. The Exponent Zero a0 = 1 for any rational number a except 0.

  12. Example 4 Simplify. a) b) c) d)

  13. Practice Simplify. 1) 2) 3)

  14. Practice Write without exponents. 1) 2) 3) Simplify. Express using exponents. 4) 5)

  15. Raising a Power to a Power For any rational number a, and any whole numbers m and n,

  16. Example 1 Simplify. Express using exponents. (52)3 = 56 (45)6 = 430 (x4)7 = x28

  17. Practice Simplify. Express using exponents. 1) (54)3 2) (22)5 3) (a6)3 4) (n4)4

  18. Example 2 Simplify. (5x)3 = (5x)(5x)(5x) = 125x3 (3z)2 = (3z)(3z) = 9z2 (2y2)4 = (2y2)(2y2)(2y2)(2y2) = 16y8

  19. Practice Simplify. 1) (3y)2 2) (6m)4 3) (2a3)3 4) (4x3)2

  20. Example 3 Simplify. (4x5y2)3 = 43x15y6 = 64x15y6 (-2x5y2)7 = -27x35y14 = -128x35y14 (2y2)4 = (2y2)(2y2)(2y2)(2y2) = 16y8

  21. Practice Simplify. 1) (4y3)4 2) (3x4y7z6)5 3) (-7x9y6)2

  22. Example 4 Simplify.

  23. Practice Simplify 3) (3x5)4 1) (34)3 2) (6x)3 5) 4) (-3m4n2)2

  24. Multiplying and Dividing Monomials

  25. Example 1 Multiply. (7y)(2y) = (7)(2)(y)(y) = 14y2 (5a3)(3a2) = (5)(3)(a3)(a2) = 15a5 (-3x3)(4xy5) = (-3)(4)(x3)(x)(y5) = -12x4y5

  26. Practice Multiply. 1) (3x)(-5) 2) (-m)(m) 3) (-x)2x3 4) (3p5q2)(4p2q3)

  27. Practice Multiply. 5) (4x5y5)(-2x6y4) 6) (-7y4)(-y)(2y3) 7) (7a5)(3a3)(-a5) 8) (9b2)(2b5)(-3b7)

  28. Example 2 Divide. 3 a 5 = 4 3 b 2 = a 2

  29. Practice Divide. 1) 2) 3) 4)

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