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121
Section 3.3
The rules that come with exponents are relatively easy to
understand, but they take some practice to ensure that you have them down
Exponents
completely. Instead of numbers we will use letters. If we multiply:
x5x8
We just have to remember what that means:
(xxxxx)(xxxxxxxx),
which is simply 13 of them multiplied together.
We write it as x13.This is our very first rule! Exponents add during
multiplication.
x5x8=x13.
The next one is quite similar:
(x5)8
Again, we just have to remember what it means:
(x5)(x5)(x5)(x5)(x5)(x5)(x5)(x5)
which is by the first rule: x40.
That gives us the second rule: Exponent to exponent will multiply.
(x5)8 = x40.
Division with exponents is just about as easy. Looking at:
This means:
and we are left with x3.
Third rule: Exponents subtract during division.
This particular rule gives rise to a couple of interesting facts. Specifically, what happens if the
top and the bottom have the same power?
= x0
But, we know that anything divided by itself is equal to 1. Thus:
x0 = 1
Secondly, what happens if the number on the bottom is larger than the one on the top.
For example:
By using the third rule we get
= x =x-3, - a negative exponent! What do we do with that?
Well, if we do it the long way, we get:
which is Section 3.3
122
Thus we have our next definition. A negative exponent puts the number on the bottom.
x-3 = Look at a couple of examples:
Using rules of exponents
23 · 22 = 25 = 2 · 2 · 2 · 2 · 2 = 32
Checking with numbers
23 · 22 = 8 · 4 = 32
= 2 = 2 = 8
=
=8
Look at that. The rules really work for any number. Here are some more examples to be able to
simplify some expressions:
(3x5)3 = 27x15 by use of the second rule.
(4y5)(7y12) = 28y17 by use of the first rule.
2-6 =
6-2 =
=
= = by the definition of a negative exponent
= 49
= = = = = Here is a summary of how you can simplify expressions with exponents:
Rule
Official
Example
Why
Multiplication –
add exponents
Exponent to a
power – multiply
exponents
aman = am+n
3x2 · 2x5 = 6x7
3xx2xxxxx = 6xxxxxxx = 6x7
Division – subtract
exponents
Exponent of 0
m n
mn
a
=a
a!
!
(a ) = a
a0 = 1 if a ≠ 0
Negative exponent
a-n =
%&
(5x ) = 125x
(5x2) (5x2) (5x2) =
125x6
36x = 9x 4x 36xxxxxxxx
= 9xx = 9x 4xxxxxx
2 3
6
= x $ by division rule
70 = 1; x0 = 1
1=
2-4 = = ;
1
= x
x 1
xx
x
=
=
= x x xxxxxxx x Section 3.3