2
–
3
1
–
n
1
–
n
1
–
3
1
–
3
1
–
2
1
–
2
1
–
2
1
–
2
1
–
2
exponential function 181
Exponential functions
9
1414
–––
–
1000
141
–
100
14
–
10
1
–
2
The multiplication rule also allows us to make sense of
fractional exponents. Consider the quantity 2 . It must
be the case that 2 ×2= 2
1= 2. Thus 2 is a value
that, when multiplied by itself, equals 2. Consequently
2= √
–
2. Similarly, 2 is a value that, when multiplied
by itself three times, equals 2, and so 2 =
3
√
—
2, the cube
root of 2. In general, 2 equals the nth root of 2. This
works for any nonzero base.
A fractional exponent indicates that a root is
to be taken: b=
n
√
—
b.
We use the power rule to make sense of other types of
fractional exponents. For example, the quantity 27
can be computed as
. In general, we have:
Finally, to compute a quantity raised to an irrational
power, one approximates the exponent by a fraction,
computes the corresponding exponential expression,
and takes the
LIMIT
as one uses better and better
approximations. For example, writing √
–
2 = 1.414…,
we see that any of the fractions 1, , , ,… can
be used to approximate √
–
2 with better and better degrees
of accuracy. We define 2√
–
2to be the limit of the values:
21= 2, , ,
, ….
The multiplication and power rules are valid even
for irrational exponents. For example, we have:
The Greek mathematician A
RCHIMEDES OF
S
YRA
-
CUSE
(287–212
B
.
C
.
E
.) was one of the first scholars to
use a special word for the power of a number. He
called the quantity 10,000, 104, a myriad, and he used
the phrase “myriad of myriads” for 10,000 squared,
104×104= 108. The ancient Greeks, for whom mathe-
matics was synonymous with geometry, called the
square of an unspecified quantity a tetragon number,
meaning a “four-corner number.” D
IOPHANTUS OF
A
LEXANDRIA
(ca. 200–284
C
.
E
.) used the Greek word
dynamis, meaning “power,” for the square of an
unknown, and called a third power a “cube,” a fourth
power a “power-power,” and fifth and sixth powers
“power-cube” and “cube-cube,” respectively.
It took many centuries for scholars to begin using
symbols to denote unknown quantities. German
mathematician Michael Stifel (ca. 1487–1567) was
the first to develop a notational system for powers of
an unspecified quantity x. He denoted the fourth
power of xsimply as xxxx. Other scholars developed
alternative notational systems. Scholars eventually
settled on the notational system French mathemati-
cian and philosopher R
ENÉ
D
ESCARTES
(1596–1650)
introduced in 1637, the one we use today. Although
Descartes considered only positive integral expo-
nents, later that century the English mathematician
S
IR
I
SAAC
N
EWTON
(1642–1727), inspired by the
work of J
OHN
W
ALLIS
(1616–1703), showed that the
same notational system can be extended to include
negative, fractional, and irrational exponents. L
EON
-
HARD
E
ULER
(1707–83) later allowed for the possibil-
ity of complex exponents.
See also
COMPLEX NUMBERS
;
EXPONENTIAL FUNC
-
TION
;
LOGARITHM
.
exponential function Any function or quantity that
varies as the power of another quantity is called expo-
nential. Precisely, if bis a positive number different
from one, then the function f(x) = bxis called the expo-
nential function with base b. The function is defined
for all real numbers x. (This would not be the case if
bwere negative: the value b, for example, would
not make sense.) The graphs of y= 2xand
y
x
=
1
2
2222
2222 2
===
×
2 2 665
1414
1000 ≈.
2 2 2 657
141
100 100 141
=
()
≈.2 2 2 639
14
10 10 14
=
()
≈.
bb b
p
qq
pqp
==
()
×
1
=
27 27 27 3
1
321
32322
==
()
=
×()