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### Exponents)

1. 1. Exponents byMaluleke Matimba Elliot
2. 2. Introduction Exponents represent repeated multiplication. For example,
3. 3. Introduction More generally, for any non-zero real number a and for any whole number n, In the exponential expression an, a is called the base and n is called the exponent.
4. 4. Exponents Are Often Used in Area Problems to Show the Feet Are SquaredLength x width = area 15ftA pool is a rectangle .Length = 30 ft. 30ftWidth = 15 ft. 2Area = 30 x 15 = 450ft.
5. 5. Exponents Are Often Used in Volume Problems to Show the Centimeters Are CubedLength x width x height =volumeA box is a rectangle 10Length = 10 cm. 10Width = 10 cm. 10Height = 20 cm. 3Volume =
6. 6.  a2 is read as ‘a squared’. a3 is read as ‘a cubed’. a4 is read as ‘a to the fourth power’. ... an is read as ‘a to the nth power’.
7. 7. Location of Exponent An exponent is a little number high and to the right of a regular or base number. 4 Exponent Base 3
8. 8. Definition of Exponent An exponent tells how many times a number is multiplied by itself. 4 Exponent Base 3
9. 9. Some Definitions of Exponents
10. 10. How to read an Exponent This exponent is read three to the fourth power. 4 Exponent Base 3
11. 11. Properties of Exponents
12. 12. Example:
13. 13. Properties of ExponentsHomework.
14. 14. Example:
15. 15. Ex: All of the properties of rational exponents apply toreal exponents as well. Lucky you!Simplify: 2 3 2 3 5 5 5 Recall the product of powers property, am an = am+n
16. 16. Exponential Functions and Their Graphs
17. 17. The exponential function f with base a isdefined by f(x) = axwhere a > 0, a 1, and x is any realnumber.For instance, f(x) = 3x and g(x) = 0.5xare exponential functions. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
18. 18. Let’s examine exponential functions. They are different than anyof the other types of functions we’ve studied because the independentvariable is in the exponent. Let’s look at the graph of this function by plotting some points. x 2x x 3 8 f x 2 8 7 6 2 4 BASE 5 4 1 2 3 0 1 Recall what a negative 2 1 -1 1/2 exponent means: -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 1/4 -2 -3 -3 1/8 1 1 -4 -5 f 1 2 -6 2 -7
19. 19. The value of f(x) = 3x when x = 2 is f(2) = 32 9 =The value of f(x) = 3x when x = –2 is 1 f(–2) = 3–2 9 =The value of g(x) = 0.5x when x = 4 is g(4) = 0.54 0.062 = 5 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
20. 20. The graph of f(x) = ax, a > 1 y 4 Range: (0, ) (0, 1) x 4 Horizontal Asymptote Domain: (– , ) y=0 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
21. 21. The graph of f(x) = ax, 0 < a < 1 y 4 Range: (0, )Horizontal Asymptotey=0 (0, 1) x 4 Domain: (– , ) Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
22. 22. Example: Sketch the graph of f(x) = 2x. yx f(x) (x, f(x))-2 ¼ (-2, ¼) 4-1 ½ (-1, ½) 2 0 1 (0, 1) 1 2 (1, 2) x 2 4 (2, 4) –2 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
23. 23. Compare the graphs 2x, 3x , and 4x f x 4xCharacteristics about the Graph of anExponential Function where a >1 x f x a1. Domain is all real numbers f x 3x2. Range is positive real numbers f x 2x3. There are no x intercepts because there is nox value that you can put in the function to make it=0 Can these exponential What is the range of of Are you they intercept What is the xthe What is see domain an intercept horizontalincreasing of these exponential or exponential function? functions asymptote an exponential of these exponential for these functions? functions? function? decreasing? functions?4. The y intercept is always (0,1) because a 0 = 15. The graph is always increasing6. The x-axis (where y = 0) is a horizontalasymptote for x -
24. 24. Exponential Equations Let a ∈ R – {–1, 0, 1}(a is a real number other than –1, 0 and 1). If am = an then m = n.
25. 25. Examples: 2x = 16 3x+1 = 81 22x + 1 = 8x – 1
26. 26. Exercises:
27. 27. References:Damirdag, M. (2011, 07 23). Power of Real numbers. Retrieved 03 20, 2013, fromSlideshare: http://www.slideshare.net/mstfdemirdag/exponents-8693171Garcia, J. (2010, July 01). Exponential Functions. Retrieved March 15, 2013, fromslideshare: http://www.slideshare.net/jessicagarcia62/exponential-functions-4772163Gautani, V. L. (2012, October 28). Multiplication properties of exponents. RetrievedMarch 17, 2013, from slideshare: http://www.slideshare.net/sirgautani/multiplication-properties-of-exponents-14917484Joshi, N. (2011, 04 01). Laws of Exponents. Retrieved 03 15, 2013, from Slideshare:http://www.slideshare.net/entranceisolutions/laws-of-exponents-7479833Yuskaits, M. (2008, 06 05). Exponents. Retrieved 03 17, 2013, from slideshare:http://www.slideshare.net/hiratufail/exponents1