Number Sense and Numeration Exponents

NUMBER SENSE AND NUMERATION: EXPONENTS Play the Exponents less than and Greater than Game 1. Evaluate each power. a. 4 2.  3 b.    5 3 4 c. − 7 1 7×7×7×7×7 e. 3 × y × y × y f. 2n × 2n × 2n × 2n × 2n h. d ×d ×d ×6×6×6×6 7×7× m× m× n 7. j. 2lk × 2lk × 3lk × 3l × 3k × 3k 8. (−5) × (−5y ) × (−5y) 5. 6. x1 = x x m × x n = x m+ n x m ÷ x n = x m− n , x ≠ 0 (x m )n = x m× n (xy) m = x m y m m  x xm  y  = y m , y ≠ 0 1 x −m = m , x ≠ 0 x Write each as a repeated multiplication. a. 9 f. 5  1 7 × −   2 b. 5s 3 4  2 g. −    3 c. (4a ) d. −12 2 p12 ÷ p 8 g. h. 7 3 y 2 (32 )4 h. (2 4 )4 −53x 4 i. Determine the value of x. a. 4 2 × 4 x = 4 6 b. 5 x × 5 3 = 5 9 m6 ÷ m x = m2 g. (4 2 )x = 4 8 j. 6  116   3  m 2 d. 12 4 ÷ 12 3 e. 2 4 ÷ 2 4 i. j. (s 3 )2 Write each expression as a simple positive power. (−5)5 × (−5)3 43 96 × 9 b. c. a. (−5)3 94 44 × 44 f.  2 e.    n 4 4 Simplify. Leave your answer in exponential form. a. 7 3 × 7 2 b. 5 × 5 6 c. z 4 × z 8 f. 6. 2. 3. 4. d. i. 5. (−4 ) 1 2×2×2 5 g. r×r×r×r 4. d. Recall some of the rules of exponents: 1. x 0 = 1, x ≠ 0 5 Write the following in exponential form. a. 11 × 11 × 11 × 11 × 11 b. 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 c. 3. 6 d. m11 ÷ m5 × m4 (−6)8 (−6)5 × (−6)5 c. t 3 × t x = t 4 d. 8 7 ÷ 8 x = 8 5 e. 7 x ÷ 7 9 = 7 h. i. j. (14 x )6 = 14 36 (b x )3 = b 3 2 2 x × 23 = 2 Expectations: i) Substitute and evaluate algebraic expressions involving exponents ii) demonstrate understanding of exponent division, multiplication, and power of a power. . 7. Evaluate all of the powers of −3 up to (−3) , beginning with (−3)1 . 10 When will (−3)n be negative? a. i. When will (−3) be positive? b. i. 8. 9. ii. Why do you think this is so? Simplify. Leave your answer in exponential form. a. 4 3 × 5 3 b. 10 8 × 38 d. 5 2 × 5 2 × 5 2 + 5 6 e. 8 4 × 34 × 2 4 Simplify. Leave your answer in exponential form. 67 10 v 43 b. c. a. 97 23 4v f. 10. ii. Why do you think this is so? n 24  22   4 2  g. 2 5a  25 a   125 a  h. c. 7 x × 8 x f. 9 t × 2 t × b t 10 9 d. 35 × 122 217  1   2 9 4a 2 × 2a 3 16 i. e. j. 62  52   2  10 7 7 × 492  21   3 4 a. On a quiz, Fauna wrote the following: 4 4 × 4 2 = 16 6 . i. What mistake did Fauna make? ii. What does 4 4 × 4 2 equal? b. On the same quiz, Mark calculated 15 3 ÷ 5 2 = 31 . i. What mistakes did Mark make? ii. Calculate 15 3 ÷ 5 2 . 11. Lani’s giant rectangular backyard is 9 5 m long and 9 3 m wide. What is the area of the backyard? Express your answer in exponential form. 12. Keiko’s gigantic cube-shaped bedroom is 7 5 mm long. What is the volume of her bedroom? Express your answer in exponential form. CHALLENGE YOURSELF! 13. Simplify the following expressions: 2 5 1 1 4a x × 2a 6 b. 12 2 × 153 × 12 4 × 159 a. 16 3 d. 5 94 × 96 2 4 93 × 95 e. xy − 7 5 1 × y −34 x 9 ( (−y −18 ) x 6 y 5 2 c. 215 × 47 × 8−22 − 7 14 )x −3 f. 11 18 7 4 × 7 −9 + 7 −5 7 −12 Expectations: i) Substitute and evaluate algebraic expressions involving exponents ii) demonstrate understanding of exponent division, multiplication, and power of a power. . EXTENSION 14. In the cross word below, solve for the value of α . (where there is an exponential solution, such as 25 , solve for the actual value). 1 2 3 4 – 5 6 . 7 . 8 9 10 11 12 – 13 14 Across 2. 59 ÷ 510 × α = 1 23 3. (65 )6 Down 1. (5 + 2)3 = α −12c (6c3 )(−3c 2 ) 3 3 92 × 37 = α 6 6 2. =α (2 × 3c5 )2 (−5pe ) ( pe) 3 2 −1 =α 4. 5d + 17c + 8d + α c = 13(d + c) 3. 5. 23.5 × 2α × 43.6 = 2048 4. −3t 5 = α3 2 24t 6. α 3 = 3581577n3 7. 3p × (951p − 183p) = α 2 10. (5u5c8 )( 0.1(102 × 103 ) (0.12 ) (104 ) 3 2 6. (10 ) (0.1 (−b−3 )5 = −bα 10 8. −3 3 7 4 3 × 0.1 )10 =α 3 2 2  7  7 9. 130   − 62   = α  2  2 (o2 m12 × m15 ) (m8 × m3n4 ) (m2 no0 )3 9 11. (2e)2 pe2 2 = mα n y o z 11. 12. −uc 6− o −4u 2 cα ) = 20u 6 c 2+ 4o 43 × 22n 164 = 2 2α ÷ 83 −11u5s7 −729s (us3 ) 3 = 11u 2 α3 8x 25 y14 z 4 (3xyz ) 2 Did You Know? There are more than 272 possible grids of classic Sudoku. 13. 62 x 4 z 6 (x10 y 8 ) 2x 2 2 =α 89 (89α + 2 k ) × 89e l 59d 3 14. l 4d +1 (8915e+3 k ) 2 = 8915e kl 67 d × 892e+1 l 12d +1 × 8943e Expectations: i) Substitute and evaluate algebraic expressions involving exponents ii) demonstrate understanding of exponent division, multiplication, and power of a power. . NUMBER SENSE AND NUMERATION: EXPONENTS Answers: 81 625 c. −117 649 d. −1024 1. a. 64 b. 2. a. 115 b. 38 c. 1 23 d. 1 75 e. 3y 3 5 r4 h. d 3 64 7 2 m2 n i. (−5)3 y 2 j. f. 3. (2n )5 g. a. 9 × 9 × 9 × 9 × 9 c. 4a × 4a  2  2  2  2  2  2 e.              n  n  n  n  n  n  2  2  2  2 g. −          3  3  3  3 i. −53× x × x × x × x b. 5 × s × s × s × s d. −12 × 12 × 12 × 12  1  1  1 f. 7  −   −   −   2  2  2 h. 7 y × 7 y × 7 j.  116   116     m × m × m  m × m × m c. z12 h. 216 d. 121 i. s 6 c. (−5)5 d. c. 1 h. 6 d. 2 i. 1 4. a. 7 5 f. p 4 b. 5 7 g. 38 5. a. 9 3 b. 6. a. 4 f. 4 b. 6 g. 4 7. −3, 9, −27, 81, −243, 729, −2187, 6561, −19683, 59049 1 45 22 × 34 l 4 k 5 e. 2 0 = 1 j. m10 1 (−6 )2 e. 10 j. −1 a. i. (−3)n will be negative when n is odd. Rule: if a pair of numbers has the same sign, then their product is positive. Otherwise, their product is negative. ii. Since there will be an odd number of − ’s, the product will be negative. b. i. (−3)n will be positive when n is even. ii. Multiplying −3 by itself an even number of times will give a product that is positive.