Download Exponents and their Properties

Exponents and
their Properties
Multiplying Powers with the
Same Bases
Since x2 = x . x, x4 = x . x . x . x, then
x2 . x4 = (x . x) . (x . x . x . x) = x6, and in
general:
The Product Rule:
For any number a and any
positive integers m and n,
am .
an
=
m6
3) (x +
1)2
1)
5) b
7
b
4
Zero as an Exponent
2)
47 .
4)
r2
7)
48
+r
3
(5x )
5x 2
8)
3x 4 y 2
6 xy
Since x2 = x . x, x4 = x . x . x . x, then
x4 x ⋅ x ⋅ x ⋅ x
=
= x2
x2
x⋅x
and in general:
am
= a m− n
n
a
am + n
(x + 1)
Dividing Powers with the Same
Bases
The Quotient Rule:
For any number a and any
positive integers m and n,
Examples: Simplify,
m3 .
Goal: Apply the properties
of exponents to simplify
algebraic expressions.
By the quotient rule
x3
= x 3− 3 = x 0
3
x
3
but simplifying x = 1 .
x3
1
Raising a Power to a Power
The Exponent Zero:
For any real number a, a ≠ 0,
a0 = 1.
Since (x3)2 = (x . x . x) . (x . x . x) =
x6 =
x3 ⋅ 2 ;
For any number a and any whole
numbers m and n,
(am)n = amn.
2 more rules:
Raising a Product to a Power:
For any numbers a and b and any
whole number n:
(ab)n = anbn
Raising a Quotient to a Power:
For any numbers a and b, b ≠ 0, and
any whole number n:
n
an
a
  = n
b
b
2