Transcript Chapter 8
Chapter 8 Exponents and Exponential Functions Alexander Townsend What Are Exponents A symbol or number placed above and after another symbol or number to denote the power to which the latter is to be raised, makes now sense right, here like this… How it works is, take the larger number on the left and multiply it by itself the number of times represented by the number on the right. Exponents can be written like you see … here → Or by using a carrot symbol 2^2, 2^4, 2^6, etc. Simple examples: 2^2=2x2 → 2x2=4 2^3=2x2x2 → 2x2x2=8 2^4=2x2x2x2→ 2x2x2x2=16 Chapter 8 Section 1 Multiplication Properties of Exponents Product of Powers: to multiply exponents that have the same base number (number on the left) you add there powers (number on the right), 5^3 x 5^4 = 5^7 → 5x5x5x5x5x5x5 = 78125 watch it’s easy… Also can be written like this… then you multiply there powers (3^4)^2 = 3^8 → 3x3x3x3x3x3x3x3 = 6561 • We can do the same thing even if we have different base numbers… (4 x 5)^2 = 4^2 x 5^2 → (4x4) x (5x5) = 16x25= 400 •You can also do the same thing with variables… (y^4)x(y^5)=(y^9) or (y^2z)^2= (y^4)(z^2) As you know, exponents can be positive numbers, negative numbers, or zero. Chapter 8 Section 2 Zero and Negative Exponents Any number to the 0 power equals 1, looks like this… 2^0=1, 3^0=1 → There all the same, they all equal 4^0=1, 5^0=1 Zero as long as the base doesn't Any negative exponent is the same as a positive exponent just with a 1 over it, you could call a^0=1, y^0=1 equal zero… 0^0 ≠ 0 • negative and positive exponents reciprocals of each other… 3^(-2) = (1/3^2) So in words: Take the base and make it a fraction by making the numerator one, then take away the negative exponent and make it positive 3^(-6)=(1/3^6)… 1/729 *Be careful negative exponents and negative bases are two different things, Shown above is the way to evaluate negative exponents, here is the way to solve negative bases,,, you know that (-)x(-)=(+), (+)x(+)=(+), and that (+)x(-)=(-) it works the same way with exponents with negative bases... (-2)^2=4 but (-2)^3=-8, if odd power answer is negative, even power means positive answer. Chapter 8 Section 3 Graphs of Exponential Functions A simple exponential function is represented by y=ab^x, just graph like any other graph, take your x coordinates make a table and find y coordinates and plot points to form graph. X -2 -1 0 1 2 3 Y= 3(1/2)^x 12 6 3 3/2 3/4 3/8 From the graph of exponential functions you can find there domain and range. This graph has a domain of x=all real #’s and a range of y=all real #’s > 0 Chapter 8 Section 4 Division Properties of Exponents You know that Multiplication and Division are opposites, so the same is true about multiplication properties of exponents and division properties of exponents, they are also opposites. For example… (6^4)x(6^2)=6^6, Multiplication Powers Property (add powers) (6^4)/(6^2)=6^2, Division Powers Property (subtract powers) More different examples… (2/3)^2= (2^2)/(3^2)= 4/9, *don’t forget to distribute the power to top and bottom. (-3/y)^3=(-3^3)/(y^3)= -27/y^3 Try to Simplify Exponential Functions that look hard… (2z^2y/3z) X (9zy^2/y^4)= (18z^3y^3)/(3zy^4) → 6z^2y^-1 → (6z^2/y) More Examples: 3^9/3^5 = 3^4 X^5/X^? = X^2 → ?=3 (1/6)^4 = 1/? → ?= 1296 *They all use the same basic rules its just to do different problems. Chapter 8 Section 5 Scientific Notation A method of expressing number in a different way, *multiplied by 10 to the appropriate power (exponent). That makes no sense right? But really its pretty easy... Rather then trying to explain and losing half of you… just watch and learn… _____________________________________________________ Scientific Notation… 2.83 x 10^1 4.9 x 10^5 8 x 10^-1 1.23 x 10^-3 → → → → Regular Notation… 28.3 490,000 0.8 0.00123 Now that we’ve tried a couple lets try to explain it… • When you see Scientific notation the first thing to do is look at the power over the 10(*when dealing in Scientific notation your base will always be 10). The power will tell you which way to move the decimal, because that is all we are doing in Scientific Notation, (Multiplying by 10^a certain number makes your original numbers decimal move either right or left a certain number of times). Here is the best and easiest way to explain it… •If the power of 10 is positive you move the decimal to the right the number of times in the power. •If the power of 10 is negative you move the decimal to the left the number of times in the power. Chapter 8 Section 6 Exponential Growth Functions * Basic Exponential Growth Function Graph Something grows exponential if is increase the same amount in every unit of time. In other words if increases at the same rate all the time. These can be tricky but with practice you’ll get them no problem. The function you use to solve them is… y=C(1+ r)^t nothing but blah,blah,blah let’s clear it up… C or P= the amount you have before growth occurs, C can be P when dealing with $ r= the growth rate t= time (the number of times the growth happens) And both C and r are positive (1+ r) is the Growth Factor → The problems usually start as a word problem, all you do is find what you need and plug it into the function and solve it is easy after awhile, Example: *Let y be the weight of the salmon during the first six weeks and let t be the number of days. The initial weight of the salmon C is 0.06. The growth rate (r) is 10%(0.10). Set it Up… y = C(1+r)^t * The only hard y = 0.06(1+ 0.10)^42 (42 days = 6 weeks) part is finding then simplify what you need y = 0.06(1.1)^42 in the question, y = 3.29 * They are all use the same formula, if you can do one you can do them all. but once you do they are easy. Chapter 8 Section 7 Exponential Decay Function These will be easy for you they are the same as Exponential Growth function except on little difference, they get solved the same way and everything there just a small difference in the formula you use… y=C(1- r)^t C or P= the amount you have before decay occurs, C can be P when dealing with $ r= the decay rate t= time (the number of times the decay happens) See the difference C and r are negative (1 - r) is the Decay Factor → *See that is the only difference they are all the same except the negative… Example: Let y be the value of the car and let t be the number of years of ownership. The initial value of the car C is $ 16,000. The decay rate r is 12%, or 0.12. y= C( 1 – r )^t =16,000 ( 1- 0.12)^t =16,000 (0.88)^t Now take the decay over 8 years, y= 16,000 (0.88)^8 = 5754 That’s a rap! Now you have the basic knowledge of exponents. You can use them on into geometry, Algebra 2, Pre Calc, Trig, and on into Calculus. More rules and new concepts will be added onto the hopefully strong foundation set down for you in this class.