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.
