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Understanding Monomials, Dividing and Simplifying Expressions

This lesson covers the basics of monomials, exponents, and simplifying expressions. It includes examples and practice problems on multiplying and dividing monomials, as well as simplifying expressions with negative exponents.

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Understanding Monomials, Dividing and Simplifying Expressions

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  1. Problems of the Day Simplify. a. – 9 + 14 e. b. 8 – 19 f. c. – 15 – (– 15) g. h. d. – 10 + (– 46)

  2. Algebra 1 ~ Chapter 8.1 Multiplying Monomials

  3. Monomials • A monomial is a number, a variable, or a product of a number and one or more variables. (ex. 4x, 3ab, y3, ) • An expression involving the division of variables is not a monomial. (ex. ) • Monomials that are real numbers are called constants. (ex. 4, −100)

  4. Exponents • An expression in the form xn is called a power and represents the product you obtain when x is used as a factor n times. • The number x is the base, and the number n is the exponent. • For example, 25 = 2•2•2•2•2 or 32

  5. Look for the pattern when multiplying powers… • 24• 23 = 2•2•2•2 • 2•2•2 or 27 • 32• 37 = 3•3 • 3•3•3•3•3•3•3 or 39 • 63• 62 = 6•6•6 • 6•6 or 65 • x5• x4 = x•x•x•x•x • x•x•x•x or x9 • am• an = am+n • To multiply two powers that have the same base, add the exponents.

  6. Example 1 – Simplify each expression a.) x3 • x5 b.) a • a6 c.) 2y3 • 4y8 • y2 d.) −3m2 • 5m4 e.) a2b2 • a3b5 = x8 = a7 = 8y13 = −15m6 = a5b7

  7. Look for the pattern when finding the power of a power… • (24)3 = 24 • 24 • 24or 212 • (32)5 = 32 • 32 • 32 • 32 • 32or 310 • (x3)2 = x3 • x3or x6 • (am)n = am • n(Power of a power) • To find the power of a power, multiply the exponents.

  8. Look for the pattern when finding a power of a product… • (2x5)2 = (2x5) • (2x5) or 4x10 • (2x5)2 = (2)2(x5)2 or 4x10 • (−3a3b)2 = (−3)2(a3)2(b)2 or 9a6b2 • (ab)m = ambm (Power of products) • To find the power or a product, find the power of each factor and multiply

  9. Example 2 – Simplify each expression a.) (x3)6 b.) (ab)3 c.) (2y3)4 d.) (−3m2n)3 e.) [(a2)3]4 = x18 = a3b3 = 16y12 = −27m6n3 = [a6]4 = a24

  10. Algebra 1 ~ Chapter 8.2 Dividing Monomials

  11. = y6 − 2 = y4

  12. Example 1: Simplify each expression. B. C. A.

  13. Example 1: Simplify each expression. E. D. = 6•x7-2 ∙ y3-1 ∙ z5-1 = 6x5y2z4

  14. Another example…

  15. Example 2 – Simplify each expression A.) Use the Power of a Quotient Property.

  16. Example 2B - Simplify

  17. Anything divided by itself is equal to 1! x2 - 5 = x-3 =

  18. Example 3: Simplify each expression. A.) 4–3 B.) 70 7º = 1 C.) (–5)–4 D.) –5–4

  19. or What if you have an expression with a negative exponent in a denominator, such as ? Definition of a negative exponent An expression that contains negative or zero exponents is not considered to be simplified. Expressions should be rewritten with only positive exponents.

  20. B.) Example 4: Simplify each expression A.) 7w–4

  21. and C.)

  22. Lesson Wrap-UpSimplify each expression. a. = y7 b. = 64d6 c. = – 32a17b17 = – 720u22v29 d.

  23. Lesson Wrap Up– Simplify each expression below. 1.) 2.) 3.)

  24. Assignment Homework • 8.1/8.2 Study Guide

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