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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 8: EXPONENTS


SECTION 8.1: THE LANGUAGE OF EXPONENTS

Exponents (or "powers") are used to simplify the multiplication of factors that are the same. For example, 2 x 2 x 2 x 2 can be written as 2 4or (7)(7)(7)(7)(7)(7) can be written as 7 6.

In 7 6, the number, 7, is called the "base" and 6 is called the "exponent" or "power." The entire structure, 7 6, is called an "exponential."

exponential


SECTION 8.2: A DEVELOPMENT ON THE POWER OF 0 AND NEGATIVE POWERS

a) Powers of 0

Let's examine the following exponentials of base 2.

exponential of base 2 21 22 23 24 25
exponential as factors of base 2 2 2x2 2x2x2 2x2x2x2 2x2x2x2x2
exponential of base 2 evaluated 2 4 8 16 32

It appears that each "exponential evaluated" can be produced from the succeeding "exponential evaluated" divided by the base, in this case 2. See the table below and make sure you see the pattern!

exponential of base 2 21 22 23 24 25
exponential as factors of base 2 2 2x2 2x2x2 2x2x2x2 2x2x2x2x2
exponential of base 2 evaluated 2 = 4/2 4 = 8/2 8 = 16/2 16 = 32/2 32


Now what if someone asked, "How would I evaluate 2 0 ?" Well, let's use the above pattern to determine the answer.

exponential of base 2 2 0 21 22 23 24 25
exponential as factors of base 2 no representation 2 2x2 2x2x2 2x2x2x2 2x2x2x2x2
exponential of base 2 evaluated 1 = (2)/2 2=(4)/2 4 = (8)/2 8 = (16)/2 16 = (32)/2 32

So, the exponential 2 0 must be 1. Why? Because I took the succeeding evaluated exponential 21 which is 2 and divided this result by 2 to get 1. Hence 2 0 must be equal to 1 based on the PATTERN that was established when we defined the entire concept of exponentials. I'm satisfied with this "procedure" for creating the answer for 2 0, but are you?

b) Neagative Powers

Now how would we deal with exponentials that have negative powers. Now what is the answer to 2 -1 ? Let's continue this "pattern rule" to determine 2 -2 and 2 -3. Consider the table below. Start from the right of the table, at 2 1 and move to the left.

exponential of base 2 2 -4 2-3 2-2 2-1 20 21
exponential of base 2 evaluated 1/16 = (1/8)/2 1/8 = (1/4)/2 1/4 = (1/2)/2 1/2 = (1)/2 1 = 2/2 2
exponential of base 2 represented as a power 1 / 24 1 / 23 1 / 22 1 / 21    

Hopefully this alternative approach to powers of 0 and negative powers has been informative.



SECTION 8.3: IMPORTANT EXPONENTIAL RULES

The following are exponential rules that you must remember and know how to use:

Rule 1) a 0 = 1 where "a" is any real number

ex. 1) 5 0 = 1     ex. 2) x 0 = 1     ex. 3) ((5)(6)(2)) 0 = 1

Rule 2) a -b =  1/ab where "a" and "b" are real numbers

ex. 1) 2 -3 = 1 /23         ex. 2) x -5 = 1/x5

Rule 3) (a b ) c = abc  where "a," "b" and "c" are real numbers

ex. 1) (23) 3 =  29        ex. 2) (x5) 3 = x 15      ex. 3) (x-5) 3 = x -15     ex. 3) (x-4) -2 = x 8

 

Rule 4) a b a c = a b+c  where "a," "b" and "c" are real numbers

ex. 1)  2325  = 28      ex. 2)  x7x5  = x12 

 

Rule 5) MATH where "a, " "b" and "c" are real numbers

 

Rule 6) MATH where "a," "b," "c, " "d" and "e" are real numbers

 

Rule 7) MATH or $\dfrac{1}{a^{c-b}}$

 

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THE SUMMER STUDY GUIDE
BY CHAPTERS

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RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE

- CHAPTER 1: THE NUMBER SYSTEM

- CHAPTER 2: ORDER OF OPERATIONS

- CHAPTER 3: VARIABLES, MONOMIALS, BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS

- CHAPTER 4: SIGNED NUMBERS, ABSOLUTE VALUE, AND INEQUALITY SYMBOLS

- CHAPTER 5: FACTORS, COMMON FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS

- CHAPTER 6: PROPERTIES OF NUMBERS

- CHAPTER 7: THE WORLD OF FRACTIONS

- CHAPTER 8: EXPONENTS

- CHAPTER 9: ROOTS

- CHAPTER 10: ALGEBRAIC EXPRESSIONS

- CHAPTER 11: CARTESIAN COORDINATE SYSTEM

- CHAPTER 12: SETS, RELATIONS AND FUNCTIONS

- CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS

- CHAPTER 14: X-INTERCEPT(ZERO) AND Y INTERCEPT(B)

- CHAPTER 15: LINES

- CHAPTER 16: FUNCTIONS

- CHAPTER 17: MULTIPLYING POLYNOMIALS

- CHAPTER 18: FACTORING

- CHAPTER 19: RATIONAL EXPRESSIONS

- CHAPTER 20: SOLVING EQUATIONS

- CHAPTER 21:SOLVING INEQUALITIES

- CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS

- CHAPTER 23: QUADRATICS

- CHAPTER 24: CIRCLES

- CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES

- CHAPTER 26: VOLUMES