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:
ã
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.