Here are the properties
of exponents that you
need for simplifying exponential
expressions.
They work for all exponents,
positive or negative,
but thinking about why
they work for positive
exponents can be very
helpful for remembering
them. For the second property
think of a
m
as m a's multiplied together
and a
n as n
a's multiplied together
and then you can see that
when you multiply these
two there are a total
of m+n a's multiplied
together. For the third
property think of a
m
and a
n the
same way, but imagine
the n a's in the denominator
canceling with n of the
a's in the numerator,
so that what is left is
n taken away from m a's
which is m-n a's. For
the fourth property you
have n groups of m a's
which altogether makes
mn a's. The fifth properties
comes from the commutative
and associative properties
of multiplication after
you write the powers as
repeated multiplication.
The sixth properties comes
from the fact that when
you multiply fractions
you multiply straight
across.
There is nothing wrong
with an answer to a simplification
problem having negative
exponents in it, but often
book problems like you
to write your final answer
with only positive exponents
to show that you know
what negative exponents
mean. My suggestion on
this is to leave this
until the very end. Before
that you can use the fact
that these properties
hold regardless of the
sign and not pay any attention
to what the exponents
mean except when you are
dealing with numbers.
Examples:
The instruction for these
problems is to simplify
and write the final answer
without any negative exponents.
Example 1:
x3x5
Solution:
x3x5=x8
Explanation:
Just add the 3 and the 5.
Example 2:
(3a4b3)(2a3b)
Solution:
(3a4b3)(2a3b)=(3)(2)a4a3b3b=6a7b4
Explanation:
What we want to do here
is get all of the numbers
together and all of the
like bases together. For
the numbers we just do the
multiplication. For the
like bases we add the exponents.
When there is no exponent
it counts as the 1 power,
so we think of b as b
1.
Example 3:
(z2)5
Solution:
(z2)5=z10
Explanation:
With a power to a power
you multiply the exponents.
Example 4:
(-2mn7p2)5
Solution:
(-2mn7p2)5=(-2)5m5(n7)5(p2)5=-32m5n35p10
Explanation:
We first bring the power
into each factor. To raise
-2 to the 5, we can just
do it, because we have a
number. Raising to the fifth
power means multiplying
it by itself 5 times, that
is -2 times -2 times -2
times -2 times -2. For the
m there is nothing to do.
For the n and p we use the
power to power rule and
multiply the powers.
Example 5:
2-4
Solution:
2-4=1/24=1/16
Explanation:
With a number we just want
to apply the definition
of a negative exponent.
Example 6:
Solution:
Explanation:
When you divide the exponents
subtract.
Example 7:
Solution:
Explanation:
Get the like bases together
and subtract the exponents.
Example 8:
Solution:
Explanation:
First we apply the division
rule, knowing that it will
work whatever kind of exponent
that you have and that 2-7
is -5. Then since we were
asked to express our final
answer in terms only of
positive exponents, we have
to use the definition of
negative exponents.
