POWER OF A QUOTIENT
The law of exponents for a power of an indicated quotient may be developed
from the following example :
Therefore,
The law is stated as follows: The power of a
quotient is equal to the quotient obtained when the
dividend and divisor are each raised to the indicated
power separately, before the division is
performed.
Practice problems. Raise each of the following expressions to the indicated
power:
Answers:
1. 3^{4} x 2^{6} = 5,184
2. 27
3. 1/125
4. [(3)^{2}]^{3} = 729
5. 25
6. 9 x 4 x 49 = 1,764
SPECIAL EXPONENTS
Thus far in this discussion of exponents, the emphasis
has been on exponents which are positive integers. There are two types of
exponents which are not positive integers,
and two which are treated as special cases
even though they may be considered as
positive integers.
ZERO AS AN EXPONENT
Zero occurs as an exponent in the answer to a
problem such as 4^{3} + 4^{3}. The law of exponents for division
states that the exponents are to be
subtracted. This is illustrated as follows:
Another way of expressing the result of dividing
4^{3} by 4^{3} is to use the fundamental axiom
which states that any number divided by itself
is 1. In order for the laws of exponents to
hold true in all cases, this must also be true when
any number raised to a power is divided by
itself. Thus, 4^{3}/4^{3} must equal 1.
Since 4^{3}/4^{3} has been shown to be equal to both
4^{0} and 1, we are forced to the conclusion that
4^{0} = 1.
By the same reasoning,
Also,
5/5 = 1
Therefore,
5^{0} = 1
Thus we see that any number divided by itself results
in a 0 exponent and has a value of 1. By
definition then, any number (other than zero) raised
to the zero power equals 1. This is further illustrated in the following
examples:
ONE AS AN EXPONENT
The number 1 arises as an exponent sometimes as a result of division. In the
example 5^{3}/5^{2} we
subtract the exponents to get
5^{32}= 5^{1}
This problem may be worked another way as follows:
Therefore,
5^{1} = 5
We conclude that any number raised to the first
power is the number itself. The exponent 1
usually is not written but is understood to exist.
NEGATIVE EXPONENTS
If the law of exponents for division is extended to include cases where the
exponent of the denominator is larger,
negative exponents arise. Thus,
Another way of expressing this problem Is as follows:
Therefore,
We conclude that a number N with a negative exponent
is equivalent to a fraction having the following
form: Its numerator is 1; its denominator is N with a positive exponent whose
absolute value is the same as the absolute value of the
original exponent. In symbols, this rule may
be stated as follows:
Also,
The following examples further illustrate the
rule:
Notice that the sign of an exponent may be changed
by merely moving the expression which contains
the exponent to the other position in the fraction.
The sign of the exponent is changed as this
move is made. For example,
Therefore,
By using the foregoing relationship, a problem such as 3+5^{4} may
be simplified as follows:
