CHAPTER4.
EXPONENTS AND OPERATION ON
EXPONENTS
4.1 Introduction: An exponent is a simple method for multiplying that number of
identical factors. Exponentiation is mathematical operation written
in the form of an where n is an integer and a is any value.
an = a x a x a x……. a
a is called the base and n
is called the exponent.
example: a=4; n=2
an = 42 = 4
x 4 = 16
an is read as a
powered n or a raised to the power n.
4.2 Integer
exponent: the exponentiation of
integers is based on basic algebra.
4.2.1 Positive
integer exponents:
n=
positive integer
a2 = a x a is called the square of a
a3 = a x a x a
is called the cube of a
The word "raised" is usually not used neither the word
"power" so 35 is
typically pronounced "three to the fifth" or "three to the five.
4.2.2 Negative integer exponents: A negative exponent means to divide by that number of
factors instead
of multiplying. So 3−3 is the same as 1/(33),
therefore in general term we can write it as
a-n =
(1/an)
example: 5-3 can be written as
5-3= (1/ 53)
= 1/(5x5x5) = 1/125
Note* n≠ 0 , n = 0 is not defined for a-n
4.2.3 Exponents
of zero and one: We know that anything to the power 0 is 1.
Proof:
Assume a0.
By the division rule
we know that,
an/an = a(n−n) = a0. (i)
But anything divided by itself is 1, so
an/an = 1.
If an/an
is equal to 1 and from (i) we have an/an equals to a0, then 1 must equal a0.
Symbolically,
an/an = a(n−n) =
a0 = 1
we created a
fraction to figure out a0 . But division by 0 is not allowed, so our
evaluation is defined for anything to the 0 power except
zero itself.
4.3 Powers:
4.3.1 Powers of ten: In the decimal
number system, integer powers of 10 are written as the digit 1 followed by a
number of zeroes, depending on the sign and magnitude of the exponent.
For example, 103
= 1000
And
10−4 = 0.0001.
Exponentiation
with base 10
is used in scientific notation to
describe large or small numbers.
For example: 3452178 can be written as
34.52178
x 105
SI
prefixes based on powers of 10 are also used to describe various
small and large quantities.
For example,
the prefix kilo
means 103 = 1000, so
1 km= 1000m.
1 kg = 1000gm
4.3.2
Powers of two: The positive powers of 2 have a great importance in computer
science because there are 2n possible values for an n-bit
variable.
Powers
of 2 are important in set theory since a set with n members has a power
set, or set of all subsets of the original set, with 2n members.
The
negative powers of 2 are commonly used, and the first two have special names: half,
and quarter.
In
the base 2 (binary) number system, integer powers of 2 are written as 1
followed or preceded by a number of zeroes determined by the sign and magnitude
of the exponent. For example, two to the power of three is written 1000 in
binary.
4.3.3 Powers of one: The
integer powers of one are one: 1n
= 1.
4.3.4 Powers of zero: If the exponent is positive, the power of zero is zero: 0n = 0, where n > 0.
If
the exponent is negative, the power of zero (0n, where n
< 0) is undefined, because division by zero is implied.
4.3.5 Powers of minus one: If n is an even integer,
then (−1)n =
1. If n is an odd integer, then (−1)n = −1. Because of this, powers of −1 are
useful for expressing alternating sequences.
4.4 Laws
of exponent:
i)
am
+n = am . an
This
identity has the consequence
ii)
am
– n ̳ am
̳ 1
a an-m
for a
≠ 0,
iii)
(am) n = am.n
Another
basic identity is
iv)
(a.b)m = am . bm