Exponents
Zero Identity: Whenever there is a nonzero base raised to the zero power, the value is always
=
a 0 1 where a ≠ 0
General Form:
one.
Fractional Exponents: For real numbers, roots can be written as fractional exponents. When
given a fractional exponent, the numerator of the fraction is the power and the denominator is
m
m
the root.
General Form: a n n a m or n a where n ≥ 2
=
2
3
Examples: 125 =
3
2
−8
2
125= 5=
25
4
3
4
=3 −8 =−
( 2) 4 =
16
Negative Exponents: If an expression has a negative exponent, move only that expression to
the other side of a fraction and make the exponent positive. [If it was in the numerator (top),
move it to the denominator (bottom). If it was in the bottom, move it to the top.]
(Coefficients must be combined as well.)
1
1
an
or
=
=
General
Form: a
1
an
a−n
−2 5
4 5
10 x z
5y z
=
Example:
−4
6y
3x 2
−n
Multiplication Rule: If multiple expressions with the same base are being multiplied, rewrite
the base and ADD the exponents. (Coefficients must be multiplied as well.)
m
n
a m+n
General Form: a ⋅ a =
(2 x 3 ) ⋅ (5 x 4 ) = 2 ⋅ 5 ⋅ x 3+ 4 = 10 x 7
2
3
2+3
5
Examples: x ⋅ x = x = x
Division Rule: If multiple expressions with the same base are being divided, subtract the
smaller exponent from the larger, and leave the same base raised to the result on the side of
the fraction that had the larger exponent. (Coefficients must be divided/reduced as well.)
m
m−n
m
a
a
a
1
General
Form:
or
=
=
an
1
a n a n−m
4 x 5 4 5− 2
2 x3
=⋅ x =
Examples:
2
2x
2
5 y4
5 1
1
= ⋅ 6−4 = 2
6
10 y
10 y
2y
Exponents raised to another exponent: When you have an exponent raised to another
exponent, you rewrite the base and MULTIPLY the exponents.
m n
mn
General Form: (a ) = a
3 4
3 4
)
x=
x12
Examples: ( x =
(−2m 4 n 2 )3 =
(−2 3)m 4 3 n 2 3 =
−8m1 2n 6
3w0 x 3 y −1 z 9 3 ⋅1 ⋅ x 3 x 2 z 9 3 x3+ 2 z 9−6 x5 z 3
=
= =
Combine the rules together:
9 x −2 y 5 ( z 2 )3
9 y 5 y1 z 6
9 y 5+1
3 y6
(w0 becomes 1; move negative exponents; multiply exponents raised to exponents; add those
that are multiplied; subtract those that are divided; reduce coefficients)