Exponent Rules
Vocabulary
Exponent - the raised number. It identifies how many times the base is to be multiplied against itself.
(e.g. 43 = 4 . 4 . 4) Also called power, “squared” if a 2, and “cubed” if a 3.
Base - the item before the exponent. It can be a letter or a number (e.g.43 base is 4, 3x2 base is x)
Coefficient - the number in front of a letter base (e.g. 3x2 coefficient is 3, in a5 it’s an assumed 1)
*** no visible exponent means an assumed exponent of 1.
*** any base to the 0 power equals 1.
If the operation is:
and you have:
then you can:
Addition/Subtraction
Examples:
same base and same
exponent (“like terms”)
3a2 + 4a2 =
6x4 + 3x3 =
x3 + 3x2 -x2 + 2x3 =
add the coefficients, without changing
the base or exponent.
7a2
6x4 + 3x3
3x3 + 2x2
Multiplication
same base
Examples:
143 (144) =
2x8 (5x3) =
xy2 . x3yz2 =
multiply the coefficients and add the
exponents
147
10x11
x4y3z2
Division
same base
Examples:
6a 5 =
3a 3
93 =
9
4 x7 y 2 =
x2 y 2 z 2
divide the coefficients and subtract
the exponents
2a2
92
4 x5
z2
Raising to a power
Examples:
exponent is outside the ( )
(34)3 =
(x5)6 =
multiply exponents
312
x30
Distribution of an
exponent to a term
Examples:
** apply to coefficients and to
assumed exponents of 1
(xy)4 =
(3m2)4 =
(4a2bc4)3 =
(2a + 3)2 =
multiply the outer exponent with each
item in the term
x4y4
34m8 = 81m8
43a6b3c12 = 64a6b3c12
(2a + 3)(2a + 3) = 4a2 + 12a + 9
**Remember, (x + y)2 = (x + y)(x + y) Why is this different from distributing exponent?
Other Exponent Forms
What if the exponent is a negative number?
This exponent form is usually introduced in Beginning Algebra:
Exponent is negative. This
Move to other side of fraction line.
Negative Exponents
doesn’t affect coefficient
Use integer rules if mult. or
sign.
division.
3
2
−
Examples:
a3
ab
=
b 2 xy 3
xy 3
4
4 x −1y −3
xy 3
(a −3b 5 c)(b 2 d −2 e −1 ) =
b7c
a 3d 2e
x −2 yz −1
=
xy −5 z −3
y6z2
x3
− 6a 2 b −3 −3
− 6 −3 a −6 b 9
)
=
2 a −3 b
2 −3 a 9 b −3
−
(
b12
9a 15
What if the exponent is a fraction?
This exponent form is usually introduced in Intermediate Algebra:
Rational Exponents
Examples:
Exponent is a fraction.
2
Numerator is power, denominator
is index. Use fraction rules if mult.
or division.
3
a3 =
1
3
1
4
( x )( x ) =
x
x
1
3
=
1
4
2
3
(m )
6
7
x
1 1
−
3 4
x
= x
4 3
−
12 12
x
a 2 or (3 a ) 2
7
12
1
12
4
7
or
or
m or
12
x 7 or (12 x ) 7
12
7
x
m 4 or
( m)
7
4
What if the exponent is a variable, like 7x ? That’s a logarithm; we’ll talk about that later….