2.3 Integer Exponents
1. Exponentiation by a positive integer
If x is a real number and m is a positive integer, then
xm = x · x · · · · · · x
Question. What do functions of the form f (x) = xm look like (for m > 0)?
• If m = 1, we have f (x) = x, which is a
• If m = 2, we have f (x) = x2 , which is a
.
.
• For m =odd integer (3, 5, 7, ...), the graphs of f (x) = xm have similar shapes:
• For m =even integer (4,6,8...), the graphs of f (x) = xm have similar shapes:
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Properties of Exponentiation
Suppose that x and y are numbers and m and n are positive integers. Then
a) xm xn = xm+n
b) (xm )n = xmn
c) xm y m = (xy)m
Proof.
a)
b)
c)
Note:
• We define x0 to equal 1 (if x 6= 0).
• We define 00 to be undefined.
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2. Exponentiation by a negative integer
If x 6= 0 and m is a positive integer, then
x−m =
1
xm
Example. 2−3 =
• For m =odd positive integer, the graphs of f (x) = x−m have similar shapes:
This general shape is called a
.
• For m =even positive integer, the graphs of f (x) = x−m have similar shapes:
Remark.
ponents.
all properties of exponentiation mentioned above hold for negative ex-
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More properties of exponents
Suppose x and y are nonzero numbers and let m and n be integers. Then
d) x0 =
xm
e) n =
x
xm
f) m =
y
Example. Write 274000 as a power of 3.
Example. Simplify
(x2 y 4 )−3
(x5 y −2 )4
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