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Section 1.6 Properties of Exponents

Section 1.6 Properties of Exponents. Why do you need to become Exponent Experts? Terms & Definitions Base , Exponent , Power x to the 5 th power x 5 = x · x · x · x · x Rules for Exponents Negative coefficients: - x 4 = -(x 4 ) but (-x) 4 = x 4

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Section 1.6 Properties of Exponents

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  1. Section 1.6Properties of Exponents • Why do you need to become Exponent Experts? • Terms & Definitions Base,Exponent,Power • x to the 5th powerx5 = x · x · x · x · x • Rules for Exponents • Negative coefficients: -x4 = -(x4) but (-x)4 = x4 • Product x3·x5 = x3+5 = x8 • Quotient x6 / x2 = x6-2 = x4 • Power (x4)3 = x4·3 = x12 • Power of Products (x6 y9)2 = x6·2y9·2= x12 y18 • Power of Quotients (x3/y5)4 = x3·4/y5·4 = x12/y20 • Zero x0 = 1 430 = 1 • Negative x-7= 1 / x7 -1 means Reciprocal • Negative Power of Quotients (x3/y5)-1 = y5/x3 1.6

  2. Product Rule • Can x2x be simplified? x3 • Can x5y6 be simplified? no, unlike bases • Can a2b7a3 be simplified? a5b7 • Can x5+x6 be simplified? no, only products 1.6

  3. Examples – Products • (-2)4 = (-2)(-2)(-2)(-2) = 16 -24 = -(2)(2)(2)(2) = -16 • x3x2x7x = x3+2+7+1= x13 • y2y5 = y7 • xxx3 = x5 • b2cb3 = b5c • x3+x = x3+x • (-5)3 = (-5)(-5)(-5) = -125 1.6

  4. The Quotient Rule 1.6

  5. Example • What if there are more on the bottom? • x2/x5 • 1/x3 1.6

  6. When an Exponent is Zero 1.6

  7. Examples – Quotient Rule • Product is addition – Quotient is subtraction • x5x2 = x5+2 = x7 x5/x2 = x5-2 = x3 • You try: • y5/y4 = y x11/x3 = x8x9/x9 = x9-9 = x0 = 1 • x4/y2 = x4/y2xy3/y= xy2 • x2/x8 = x2-8 = x-6 = 1/x6 1.6

  8. Negative Exponents 1.6

  9. Examples – Zero and Negative • x3 = xxx x2 = xx x1 = x x0 = 1 • Think: Only the coefficient remains • 60 = 1 2y0 = 2 (3y2z)0 = 1 (x+3)0 = 1 -y0 = -1 • A negative exponent means make it the reciprocal • 6-1 = 1/6 2y-1 = 2/y (3y2)-1 = 1/(3y2) -y-1 = -1/y • 2-3 = 1/23 = 1/8 (x+3)-2 = 1/(x+3)2 • (3/7)-1 = 7/3 (x/3)-2 = (3/x)2 = 9/x2 • x-3/ x-7 = x-3-(-7) = x-3+7 = x4 1.6

  10. The Power Rule 1.6

  11. The Power Rule for Products & Quotients 1.6

  12. Examples –Powers • (y2)5 = y10 • (x2y)3 = x6y3 • (bb2b3)4 = b24 • (2x4)3 = (2x4)(2x4)(2x4)= 23x4·3 = 8x12 • (-2x4)3 = (-2x4)(-2x4)(-2x4)= (-2)3x4·3 = -8x12 • (⅓a3b)2 = (⅓a3b)(⅓a3b)=(⅓)2a3·2b1·2 = (a6b2)/9 • -(⅓a3b)2 = -(⅓a3b)(⅓a3b)=-(⅓)2a3·2b1·2 = -(a6b2)/9 1.6

  13. Serious Examples • Simplifying inside Using exponent ops 1.6

  14. Next Time … • 1.7 Scientific Notation and • 2.1 Graphs 1.6

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