Download CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The

CHAPTER 3: EXPONENTS AND POWER FUNCTIONS
1. The algebra of exponents
1.1. Natural Number Powers. It is easy to say what is meant by bn –
‘b (raised to) to the (power) n – when n is a natural number. For example:
23 = 2 · 2 · 2 = 8
34 = 3 · 3 · 3 · 3 = 81
In general, when n is a natural number and b is any real number,
bn = b| · b{z· · · }b
n
Terminology: In the expression
be
e is said to be the exponent and b the base.
1.2. The First Law of Exponents. Note what happens to the exponents
if we multiply two powers with the same base. For example:
b3 · b4 = (b · b · b) · (b · b · b · b) = b7 = b3+4
In general,
be1 · be2 = be1 +e2 (The First Law of Exponents)
Thus, when we multiply two powers with the same base we must add the
exponents.
1.3. Zero and negative powers.
Notation: b0 = 1 for any b ∈ R, 6= 0.
Example 1.1. 100 = 1, 1870 = 1, (−π)0 = 1, (1.0529)0 = 1.
Why is this the correct meaning to give to the symbol b0 ? We would like
the First Law of Exponents to continue to remain true when one or both
of the exponents are equal to 0. But then, for any b 6= 0, we must have:
b0 · be = b0+e = be = 1 · be and thus (cancelling be on both sides) we are
forced to conclude that b0 = 1.
What about assigning a meaning to b−e ? Once again the definition is forced
on us (see below) if we want the First Law of Exponents to continue to hold:
Notation:
1
(b 6= 0)
b−e = e
b
1
2
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Example 1.2.
2−3 =
1
1
=
3
2
8
Example 1.3.
−1
1
1
=
=3
3
1/3
[ Why is this the correct definition?
1
b−e · be = b−e+e = b0 = 1 = e · be . Thus, cancelling be on both sides again,
b
gives b−e = 1/be .]
1.4. The Second Law of Exponents.
Note what happens to the exponent when we raise a number to a power
and then raise this new number to another power. For example:
(b2 )3 = b2 · b2 · b2 = (b · b) · (b · b) · (b · b) = b6 = b2·3
In general we have:
(be )d = be·d (The Second Law of Exponents)
Thus when we take a power of a power, we must multiply the exponents.
√
1.5. Fractional Powers. Recall that if b > 0, then b denotes the unique
positive number √
whose square is b.
More generally, m b denotes the unique positive number whose mth power
is b.
√
√
√
Example 1.4. 3 27 = 3. 5 32 = 2. 4 81 = 3, and so on.
√
Note that m b is usually only
√ defined if b ≥ 0. However, when m is an
odd natural number, then m b makes sense for any real number b, negative,
positive or zero. For odd m, any negative number b has a unique negative
mth root. Indeed, when m is odd, and b is negative we have:
√
√
m
m
b = − −b (Here −b is positive.)
√
√
√
Example 1.5. 3 −8 = − 3 8 = −2. 5 −32 = −2, and so on.
√
Notation: b1/m = m b if b ≥ 0.
[Why is this the correct meaning for b1/m ? This time we find that this definition is forced on us by the requirement that the Second Law of Exponents
continue to hold even when one of the exponents is a fraction:
For b > 0, we have
1 m
1
bm
= b m ·m = b1 = b
and thus b1/m must be the mth root of b.]
Example 1.6.
1
83 =
√
3
8 = 2,
1
81 4 =
√
4
81 = 3
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Example 1.7.
1
1
=
251/2
5
b > 0,
1
1
= 1/m = b− m
b
1
25− 2 =
Example 1.8. For any number
1
√
m
b
1.6. Rational Exponents. Recall that the set of rational numbers, Q,
consists of those real numbers which can be expressed as one integer (whole
number) divided by another. In particular, this set includes any number such as 123.456789 - which has a finite√decimal
√ expansion.
Many real numbers - for example, π, 2, 3 - are not rational. But real
numbers can always √
be closely approximated by rational numbers. For example, π ≈ 3.14159, 2 ≈ 1.4142 etc.
We can now make sense of bq where b > 0 and q is any rational number.
For if q ∈ Q, then q = n/m for some whole numbers n and m (we can also
suppose that m > 0).
If the Second Law of Exponents is to continue to hold, we must have
√ n
m
q
n/m
1/m n
b =b
= b
=
b
√
m
q
b)n .
Notation: If b > 0 and if q ∈ Q, q = n/m
then
b
=
(
√
m
1/m
Note: This is the same as (bn )
= bn .
Example 1.9.
1 2
2
8 3 = 8 3 = 22 = 4
Example 1.10.
1 −5
5
1
4− 2 = 4 2
= 2−5 =
32
Example 1.11.
1 5
5
81 4 = 81 4 = 35 = 243
1.7. Real number exponents. We have now given a meaning to the notation bq for any positive number b and any rational number q, and in such
a way that the two laws of exponents hold for rational powers.
How can we give a meaning to the expression
br if√ r is an irrational real
√
3
number? What should be the meaning of 5 2 or π 7 , for instance?
This is a much more difficult problem. To deal properly with it would involve
a deeper understanding of the real line at the level of a more advanced
Mathematics course.
I will simply state, without proof, the following fact:
It is possible to give a definite meaning to br for any b > 0 and any real
number r – rational or irrational – in one and only one way so that
(1) It has the meaning already assigned above if r ∈ Q.
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(2) The two ‘laws of exponents’ continue to hold for all real exponents.
(3) For fixed b > 0, the function f : R → R, f (x) = bx is continuous.
We will deal with the notion of a continuous function later. For the present,
it is enough to say that this last condition means that the graph of the
function f (x) = bx is an unbroken curve and that if r is a real number and
q is a good rational approximation to r, then bq is a good approximation to
br .
Note that any scientific calculator has a y x button, which will calculate
ar where a is positive and r is any real number which the calculator can
accept as input. Of course, most of the time the calculator will only return
a finite decimal (and hence rational) approximation of the true value.
2. Power Functions
Definition 2.1. A power function is a function of the form f (x) = xr for
some fixed real number r. Note that, for positive r, the domain of this
function consists of all non-negative real numbers (this set can be written
‘[0, ∞)’), unless r is a rational number with odd denominator. (When r is
negative, 0 does not lie in the domain – why?)
Example 2.1. The functions f (x) = x, f (x) = x2 , f (x) = x3 , etc. – whose
graphs we have already considered – are power functions.
Example 2.2. The function f (x) = 1/x is a power function; it is just
the function f (x) = x−1 (so r = −1 here). The domain is the set of
all real numbers other than 0 (which can be written as ‘R \ {0}’ or as
‘(−∞, 0) ∪ (0, ∞)’). Its graph is a hyperbola:
Example 2.3. We saw that the pressure P of a fixed amount of a gas at
fixed temperature in a container varies with the volume according to the
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formula
P =
K
= KV −1 (K a constant).
V
Thus the graph of P against V has essentially the same shape as the graph
y = 1/x = x−1 .
Example 2.4. More generally, the function f (x) = 1/xm (m ∈ N) is just
the power function f (x) = x−m (with r = −m). Its domain is (−∞, 0) ∪
(0, ∞).
For example, here are the graphs of f (x) = 1/x2 , f (x) = 1/x3 and f (x) =
1/x4 (which is which?):
Example 2.5. The gravitational attraction between two large bodies (eg.
planets) follows an inverse square law ; the force of attraction F is inversely
proportional to the square of the distance r between the bodies:
F =
K
= Kr−2 (K a constant).
r2
Thus the graph of F as a function of r has essentially the same shape as
the graph of y = 1/x2 = x−2 .
Example 2.6. The function f (x) =
x1/2 . The domain is [0, ∞).
√
x is just the power function f (x) =
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√
Example 2.7. More generally, the function f (x) = m x is just the power
function f (x) = x1/m . Its domain is [0, ∞) if m is even and R if m is odd.
Here are the graphs of f (x) = x1/2 and f (x) = x1/3 (which is which?):
Example 2.8. The kinetic theory of gases tells us that the root mean square
speed, c, of molecules in a gas is given by the formula
r
c=
3RT
M
where T is the temperature (in Kelvin), R = 8.3145JK−1 mol−1 is the gas
constant and M is the molar mass of the gas. The molar mass of oxygen
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(O2 ) is 0.032 kg·mol−1 . Thus, for oxygen,
r
√
3 × 8.3145 × T
= 779.5T (in metres per second).
c=
0.032
Thus the graph
√ of c as a function of T has essentially the same shape as the
graph y = x = x1/2 .
√
Example 2.9. The function f (x) = 1/ m x is just the power function
f (x) = x−1/m .