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^{ 4}or (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."
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 
2^{1 } 
2^{2} 
2^{3} 
2^{4} 
2^{5} 
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 
2^{1 } 
2^{2} 
2^{3} 
2^{4} 
2^{5} 
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} 
2^{1 } 
2^{2} 
2^{3} 
2^{4} 
2^{5} 
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 2^{1} 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} 
2^{0} 
2^{1} 
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 / 2^{4 } 
1 / 2^{3 } 
1 / 2^{2} 
1 / 2^{1 } 


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/a^{b } where "a" and "b" are real numbers
ex. 1) 2^{ 3} = 1 /2^{3 } ex. 2) x^{ 5 }= 1/x^{5 }
Rule 3) (a^{ b} )^{ c} = a^{bc } where "a," "b" and "c" are real numbers
ex. 1) (2^{3})^{ 3} = 2^{9} ex. 2) (x^{5})^{ 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) 2^{3}2^{5} = 2^{8} ex. 2) x^{7}x^{5} = x^{12}
Rule 5) where "a, " "b" and "c" are real numbers
Rule 6) where "a," "b," "c, " "d" and "e" are real numbers
Rule 7) or



THE SUMMER STUDY GUIDE
BY CHAPTERS

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: XINTERCEPT(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
