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Exponents and Polynomials

Chapter 3. Exponents and Polynomials. Exponents. 3.1. Exponents. Exponents that are natural numbers are shorthand notation for repeating factors. 3 4 = 3 • 3 • 3 • 3 3 is the base 4 is the exponent (also called power )

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Exponents and Polynomials

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  1. Chapter 3 Exponents and Polynomials

  2. Exponents 3.1

  3. Exponents Exponents that are natural numbers are shorthand notation for repeating factors. 34 = 3 • 3 • 3 • 3 3 is the base 4 is the exponent (also called power) Note by the order of operations that exponents are calculated before other operations.

  4. Evaluating Exponential Expressions Example Evaluate each of the following expressions. 34 = 3 • 3 • 3 • 3 = 81 (–5)2 = (– 5)(–5) = 25 –62 = – (6)(6) = –36 (2 • 4)3 = (2 • 4)(2 • 4)(2 • 4) = 8 • 8 • 8 = 512 3 • 42 = 3 • 4 • 4 = 48

  5. Evaluating Exponential Expressions Example Evaluate each of the following expressions. a.) Find 3x2 when x = 5. 3x2 = 3(5)2 = 3(5 · 5) = 3 · 25 = 75 b.) Find –2x2 when x = –1. –2x2 = –2(–1)2 = –2(–1)(–1) = –2(1) = –2

  6. The Product Rule If m and n are positive integers and a is a real number, then am·an = am+n For example, 32 · 34 = 32+4 = 36 x4 ·x5 = x4+5 = x9 z3 ·z2 · z5 = z3+2+5 = z10 (3y2)(– 4y4) = 3 · y2 (– 4) · y4 = – 12y6 = 3(– 4)(y2 · y4)

  7. Helpful Hint Add exponents. Don’t forget that 35∙ 37= 912 Common base not kept. 35∙ 37= 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 7 factors of 3. 5 factors of 3. = 312 12 factors of 3, not 9. In other words, to multiply two exponential expressions with the same base, we keep the base and add the exponents. We call this simplifyingthe exponential expression.

  8. Helpful Hint Don’t forget that if no exponent is written, it is assumed to be 1.

  9. The Power Rule If m and n are positive integers and a is a real number, then (am)n = amn For example, (23)3 = 23·3 = 29 (x4)2 = x4·2 = x8

  10. The Power of a Product Rule If n is a positive integer and a and b are real numbers, then (ab)n = an·bn For example, (5x2y)3 = 53 · (x2)3 · y3 = 125x6 y3

  11. The Power of a Quotient Rule If n is a positive integer and a and c are real numbers, then For example,

  12. The Quotient Rule Group common bases together. If m and n are positive integers and a is a real number, then For example,

  13. Zero Exponent a0 = 1, a≠ 0 Note: 00 is undefined. For example, 50 = 1 (xyz3)0 = 1 · 1 · 1 = 1 = x0· y0 · (z3)0 –x0 = –(x0) = – 1

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