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CHAPTER 3 EXPONENTS When numbers are repeatedly multiplied together, we use exponent to illustrate this process. power, index, or exponent 3 x 3 x 3 x 3 x 3 = 35 base If n is a positive integer, then an is the product of n factors of a. an = a x a x a x a x . . . x a n factors We say that a is the base, and n is the exponent or index. Negative Bases A negative base raised to an odd exponent is negative. A negative base raised to an even exponent is positive. Exercise 3A 3. Simplify a. (-1)5 i. -25 k. (-5)4 5. Use your calculator to find the values of the following: a. 9-1 f. 1/34 h. (0.366)0 a m a m a n a a mn a n mn ,a 0 a a ab n a b n m a b 0 a 1 a n n a n b n n n 1 a n a n To divide numbers with the same base, keep the base and subtract the exponents. When raising a power to a power, keep the base and multiply the exponents. The power of a product is the product of the powers. ,b 0 1, a 0 n a mn To multiply numbers with the same base, keep the base and add the exponents. The power of a quotient is the quotient of the powers. Any non-zero number raised to the power of zero is 1. and in particular a 1 1 a ,a 0 Exercise 3B 1. Simplify using the laws of exponents: a. 54 x 57 c. k 8 k 3 i. (5t)3 2. Write as power of 2 a. 4 e. 32 j. 1/64 3. Write as power of 3 h. 1/81 j. 243 4. Write as a single power of 2: d. (2x+1)2 2 g. 2 m m 5. Write as a single power of 3: i. 9 3 a 1 a 7. Write the following in simplest form, without brackets: g. 2 3p 3 q 2 4a b h. 3 2 8. Write without negative exponents: a 2 b 1 f. 2 c 9. Write in non-fractional form: e. a a n 2 n Rational exponents 1 8. n an n a a where reads “the nth root of a”, nєZ. For example: 1 a2 a 1 a3 3 a Rational exponents m a n n a m for a > 0, n є Z+ , m є Z Exercise 3C 1. Write as a single power of 2: a. 1 5 f. 2 x i. 2 3 1 3 16 2 3. Write the following in the form ax where a is a prime number and x is rational: g. 1 4 27 The Expansion Laws a ( b c ) ab ac ( a b )( c d ) ac ad bc bd ( a b )( a b ) a b 2 2 ( a b ) a 2 ab b 2 ( a b ) a 2 ab b 2 2 2 2 2 Exercise 3D.1 1. Expand and simplify: e. 3 x ( 2 3 x ) 1 3 1 f. x 2 ( x 2 2 x 2 3 x 1 2 ) 2. Expand and simplify: b. (3 2)(3 x 1 i. ( x 2 x x 5) 1 1 2 )( x 2 x 1 2 ) Exercise 3D.2 1. Factorize: a. 5 e. 6 2x n2 5 -6 x 2. Factorize: e. 9x – 4x f. 4x + 6(2x) + 9 i. 25x – 4(5x) + 4 3. Factorize f. 49x – 7x+1 + 12 5. Simplify 6 2 m a. f. 2 5 n 1 m m 5 4 n 6. Simplify a. b. Exponential equation An exponential equation is an equation in which the unknown occurs as part of the index or exponent. For example: 2x = 8 and 30 x 3x = 7 Exponential equation If the base numbers are the same, we can equate exponents. If ax = ak then x = k. Exercise 3E 1. Solve for x a. 2x = 8 f. 2 x j. 3 x 1 2 1 27 2. Solve for x e. 27 g. x2 n. x 1 9 4 1 4 128 1 x 32 3. Solve for x, if possible: a. 4 2 x 1 c. 2 8 x 8 1 x 1 x 1 4 5. Solve for x: a. 4 6 ( 2 ) 8 0 x x e. 25 23 ( 5 ) 50 0 x x Exponential Functions 16.The most simple general exponential function has the form y = bx where b > 0, b ≠1. 17. y = 2x y = 2x is ‘asymptotic to the x-axis’ or ‘y = 0 is a horizontal asymptote’. 18. For the general exponential function y = a • bx-c + d where b > 0, b ≠ 1, a ≠ 0: b controls how steeply the graph increases or decreases c controls horizontal translation d controls vertical translation the equation of the horizontal asymptote is y = d if a > 0, b > 1 the function is increasing. if a < 0, b > 1 the function is decreasing. if a > 0, 0 < b < 1 the function is decreasing. if a < 0, 0 < b < 1 the function is increasing. Exercise 3F 1. Use the graph above to estimate the value of: a. 21/2 or √2 c. 21.5 6. For each of the functions below: i. sketch the graph of the function ii. state the domain and range iii. use your calculator to find the value of y when x = √2 iv. discuss the behavior of y as x∞ a. y = 2x + 1 b. y = 2 – 2x Growth and Decay 20. Exponential functions are often used to model growth and decay. Populations of animals, people, and bacteria usually grow in an exponential way. Radioactive substances, and items that depreciate in value, usually decay exponentially. Growth 21. Consider a population of 100 mice which under favorable conditions is increasing by 20% each week. Determine the exponential function for this situation. Week Population 0 P0 = 100 1 P1 = P0 x 1.2 = 100 x 1.2 2 P2 = P1 x 1.2 = 100 x (1.2)2 3 P3 = P2 x 1.2 = 100 x (1.2)3 4 P4 = P3 x 1.2 = 100 x (1.2)4 and so on. Pn = 100 x (1.2)n Exercise 3G.1 1. The weight Wt of bacteria in a culture t hours after establishment is given by Wt = 100 x 20.1t grams. a. Find the initial weight. b. Find the weight after: i. 4 hours ii. 10 hours iii. 24 hours. c. Sketch the graph of Wt against t using the results of a and b only. d. Use technology to graph Y1 = 100 x 20.1x and check your answers to a, b, and c. 3. A species of bear is introduced to a large island off Alaska where previously there were no bears. 6 pairs of bears were introduced in 1998. It is expected that the population will increase according to Bt = B0 x 20.18t where t is the time since the introduction. a. Find B0. b. Find the expected bear population in 2018. c. Find the expected percentage increase from 2008 to 2018. 23. Consider a radioactive substance with original weight 20 grams. It decays or reduces by 5% each year. The multiplier for this is 95% or 0.95. Year Weight (grams) 0 W0 = 20 1 W1 = W0 x 0.95 = 20 x 0.95 2 W2 = W1 x 0.95 = 20 x (0.95)2 3 W3 = W2 x 0.95 = 20 x (0.95)3 4 W4 = W3 x 0.95 = 20 x (0.95)4 and so on. Wn = 20 x (0.95)n Exercise 3G.2 1. The weight of a radioactive substance t years after being set aside is given by W(t) = 250 x (0.998)t grams. a How much radioactive substance was initially set aside? b. Determine the weight of the substance after: i. 400 years ii. 800 years iii. 1200 years. c. Sketch the graph of W(t) for t > 0 using a and b only. d. Use your graph or graphics calculator to find how long it takes for the substance to decay to 125 grams. The Natural Exponential ex 25. What is e? It is a mathematical constant that has a value of 2.71828182845904523536028747135266249775724709369995 It is called the Euler Number after a Swiss mathematician named Leonhard Euler. Exercise 3H 3. For the general exponential function y = aekx, what is the y-intercept? 4. Consider y = 2ex. a Explain why y can never be < 0. b Find y if: i. x = -20 ii. x = 20. 9. On the same set of axes, sketch and clearly label the graphs of: f : x ex, g : x ex-2, h : x ex+3 State the domain and range of each function. 12. The weight of bacteria in a culture is given by W(t) = 2et/2 grams where t is the time in hours after the culture was set to grow. a Find the weight of the culture when: i. t = 0 ii. t = 30 min iii. t = 1½ hours iv. t = 6 hours. b Use a to sketch the graph of W(t) = 2et/2 . 14. The current flowing in an electrical circuit t seconds after it is switched off is given by I(t) = 75e-0.15t a. What current is still flowing in the circuit after: i. 1 second ii. 10 seconds? b Use your graphics calculator to sketch I(t) = 75e-0.15t and I = 1. c Hence find how long it will take for the current to fall to 1 amp.