Luttrell 2012
100
7 - Rules of Exponents
The operation of raising a number to a power is e
x
ponentiation. In the e
x
pression
x
3
, 3
x
’s are
being multiplied together so that
x
3
=
x
∙
x
∙
x
. For the e
x
pression
x
n
,
x
is called the base, n is the
e
x
ponent and the whole e
x
pression is a power.
Remember that e
x
ponents occur second in the order of operations. So that 4
x
3
does not mean
4
x
×4
x
×4
x
, rather 4×
x
×
x
×
x
. One of the common mistakes when simplif
y
ing -2
4
is to wrongl
y
use
the negative. The expression -2
4
means the negative of 2 to the fourth or the negative of 16
which is -16. If it is -2 that is raised to the fourth power, then it needs to be written as (-2)
4
.
The properties of E
x
ponents:
1. Product of two powers with equal bases:
x
x
x
a
b
a b
⋅
=
+
2. Quotient of two powers with equal bases:
x
x
x
a
b
a b
=
−
3. Power of a power:
(
)
x
x
a
b
ab
=
4. Power of a product:
(
)
xy
x y
a
a
a
=
5. Power of a quotient:
( )
x
y
a
a
a
x
y
=
6.
x
0 = 1, as long as
x
≠0.
7.
x
x
a
a
−
=
1
8.
x
x
a
b
a
b
=
Reminders:
1. Never confuse distribution of e
x
ponents - ONL
Y
distribute over multiplication and division,
never over addition or subtraction. e.g. (x
2
+y
3
)
2
≠ x
4
+y
6
but (x
2
y
3
)
2
= x
4
y
6
.
2. The product of two fractions is made from the product of the numerators over the product of
the denominators.
e.g. (2∕3) (5∕6) = 10∕18
E
x
ample a:
x y
y
x y
2
3
5
2
8
•
=
E
x
ample b:
x
y
xy
xy
x x y
y xy
x y
xy
x
y
x y
x
y
2
3
2
3
2
2
2
3
3
4
2
6
4 1
2 6
3
4
3
4
•
=
=
=
=
=
−
−
−
(
)
