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Mastering Exponents: Rules, Examples, and Techniques

Learn why becoming an exponent expert is vital. Understand properties, rules, and examples of exponents, including product, quotient, negative exponents, and the power rule. Simplify expressions efficiently with practical examples.

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Mastering Exponents: Rules, Examples, and Techniques

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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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