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6.3 Division Properties of Exponents When working with division problems involving exponents, you can expand and cancel. 1 1 1 x/ x/ xxx x5 / 3 /x /x x/ x x2 You can cancel because any number or variable divided by itself is equal to one. Can you expand and cancel this in your head? When you are expanding and canceling the exponents in your head, what mathematical operation are you using? Subtraction When working with division problems involving exponents, you can expand and cancel. 1 1 111 3 4 3 xxxyyyy // // / /// 9x 7 y 3 / 3xxxxxxxy / 3x y y3 3x 4 Once you have practiced this, you may expand and cancel in your head! Here’s how… Expand and cancel in your head! 3x3 y 4 1 y3 4 7 9x y 3 x y3 3x 4 What is left and where is it located? Example 1 Simplify. 1. Write problem. 39 4 3 35 81 2. Expand and cancel (you may do this in your head). What is left and where is it located (numerator or denominator)? You cannot 3. Simplify. cancel the base! Example 1 Simplify. You cannot cancel the bases to one when the powers are different! 19 9 3 1 5 5 3 1 14 1 Wrong answer! Example 2 Simplify. x4 1 3 7 x x What is left and where is it located (numerator or denominator)? You cannot have a denominator without a numerator! Example 3 Simplify. 12 m5 4m 4 1 3m 4m 4 What is left and where is it located (numerator or denominator)? A nonzero number to the zero power is 1. 258 8 8 25 258 250 1 x5 55 x x5 x0 1 Zero to the zero power is undefined! Example 4 Simplify. 25x3 3 5 1 5x 5 5 1 3 25x 5 3 5x Example 5 Simplify. 53 y 6 5y 6 521 25 What is left and where is it located (numerator or denominator)? Example 6 Simplify. 83 x 4 y3 1 x2 3 2 7 8 x y y4 x2 4 y What is left and where is it located (numerator or denominator)? Power of a Quotient Property To find a power of a quotient, find the power of the numerator and the power of the denominator and then 2 divide. 4 4 2 2 2 2 2x 2 x 2 4 4 4 w w y2 5 52 y 16 16 22 x2 4 2 25 w y 4 x2 2 You must use You must use y parentheses! parentheses! Example 7 Simplify. 2 42 4 2 x x The exponent applies only to the base! 16 2 x Example 8 Simplify. 2 3 3 2 3x 3x y y3 33 x23 y3 Must use parentheses! 27x6 3 y Simplify. Change the position of all negative exponents – keep the positive powers where they are given. 77 a 5b b 5 a 3 44 a bb b11 8 a Example 9 Simplify. x 77yy66 x 22 2 x y x y8 x9 Change the position of all negative exponents – keep the positive powers where they are given. Example 10 Simplify. 3 6 4x3 yy6z Change the position of 4x 4 2 4 all negative exponents – 32xy z keep the positive powers 1 4 1 z where they are given. 8 x13y26 z5 8x 4 y8 Expressions Involving Fractions Change the position of 22 3 x3 3x x3 x all negative exponents – 3 55 55 3 x x y 2y 2x keep the positive powers xyy where they are given. 3 5 5 3x y Multiply across the top (numerators) 1 2 3 2x y and across the bottom (denominators). Expand and cancel. What is left and where is it located? 3x8 y5 2x3 y3 3 x5 y 2 2 3x5 y2 2 Example 11 Simplify. 3x3y 12x2 y2 36x5 y3 3 4x y 4x y3 9 x4 What is left and where is it located? When you are expanding and canceling the exponents in your head, what mathematical operation are you using? Subtraction Example 12 Simplify. 3 y2 y y2 y33 3 4 33 12 x xx xx 12 x Change the position of all negative exponents – keep the positive powers where they are given. y2 x15 y2 y3 x15 y 8-A3 Pages 465-466 # 26–46 even, 49-57 all. Algebra rocks! Which expressions are equivalent to 5-4 · 53 A. 5-8 5-8 · 53 B. -4 5 5-8 · 52 C. 5-3 5-8 · 54 D. 5-3 5-3 · 57 E. 53 5-4 · 52 F. 53 1 5 : You can cancel to one when the value in the numerator and the denominator are the same. 34 1 4 3 3 1 3 However, when the exponents are not the same you cannot cancel the bases. The different exponents simplify to different numbers. 34 81 3 27 3 3