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."
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 |
|
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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)  where "a, " "b" and "c" are real numbers
Rule 6)  where "a," "b," "c, " "d" and "e" are real numbers
Rule 7)  or 
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