Download Lecture 14: Oct. 30

Lecture 14: Exponents, Roots, and Logarithms
We defined exponents early in the term as
repeated products.
incrementing n times: adding n
INC(INC(INC(5)) = 5+3 = 8
adding n times:
multiplying by n
5+5+5 = 3x5 = 15
multiplying n times: raising to the nth power.
5x5x5 = 53 = 125
We can interpret exponents in terms of sets:
if n(A) = a, n(B) = b , then the number of functions
from B to A is ab.
the number of ordered b-tuples (a1,a2,...,ab) is ab.
EXAMPLE: the number of three-digit numbers you
can make using the digits 1,2,3,4,5 (repetition
allowed) is 53 .
EXAMPLE: A bit in a computer circuit can have 2
values, 1 or 0. The extended ASCII code uses 8
bits; it can store 28 = 256 different values. The
more modern international UNICODE character
set has 16 bits and can store 216 = 65,536 values.
But we haven’t done so much with them! Why?
*Exponentiation is NOT commutative.
23 = 2x2x2 = 8
32 = 3x3 = 9
*Exponentiation is NOT associative.
(ab)c = a(bc)
which is usually a lot smaller than a
(bc)
WARNING: The usual “test-an-easy-case” rule
here is tricky. The numbers get big so fast that it
is tempting to let most of the numbers be 2. But as
(22)
22 = 2x2, (22)2 does equal 2.
We can define (for instance) 3 x (4/5) by
repeated addition:
3x (4/5) = (4/5)+(4/5)+(4/5)
and because it’s commutative that also defines
(4/5) x 3
which would otherwise be a more complicated
concept, involving fractions of something other
than a whole.
We can define (for instance) (-3)5 within the
integers by repeated multiplication:
(-3)5 = (-3)(-3)(-3)(-3)(-3) = -243
But as exponentiation isn’t commutative that
doesn’t tell us anything about
5-3
which doesn’t exist within the integers.
We can define (4/5)3 by repeated multiplication:
(4/5)3 = (4/5) x (4/5) x (4/5) = 64/125
but because exponentiation is not commutative this
tells us nothing about 3 4/5 .
In fact, 34/5 (like 21/2) is not a rational number.
The rational numbers are closed under “raising to
integer powers” (except for 0a when a0)
But they are not closed under exponentiation
in general.
RULE: If an integer raised to a positive rational
power is not an integer, then it isn’t a rational
number either. (PROOF OMITTED!)
SIMPLE EXAMPLE: As 2 isn’t an integer,
by the rule it isn’t rational.
NON-SIMPLE EXAMPLE: 325/7 is not an
integer. 117 = 19487171
325 = 33554432
127 = 35831808
so 325/7 is between 11 and 12. As 325/7 isn’t an
integer, it isn’t rational.
On the other hand, the positive real numbers ARE
closed under exponentiation (even with zero or
negative exponent.)
RULES:
a0 = 1 for all a>0
(ab)c = abc
(ab)c = ac bc
ab+c = ab ac
WHAT DOES a-1 equal?
2-1 x 2
=
2-1 x 21 =
so 2-1 must be
3-1 x 3
=
3-1 x 31 =
so 3-1 must be
a-1 x a
=
a-1 x a1 =
so a-1 must be
What are:
(1/2)-1
2-2
(1/3)-2
2-3
(2/5)-1
WHAT MUST a1/2 be?
41/2 x 41/2 =
so 41/2 must be
31/2 x 31/2 =
so 31/2 must be
a1/2 =
161/2 =
(1/4)1/2=
4-1/2 =
(1/4)-1/2 =
(4/9)1/2=
(9/4)1/2 =
(9/16)-1/2=
WHAT MUST am/n be?
a1/2 =
81/3 x 81/3 x 81/3 =
81/3 =
a1/3 =
a1/4 =
a1/n =
a1/3 x a1/3 =
82/3 =
a2/3 =
272/3 =
642/3 =
84/3 =
43/2 =
1005/2 =
What is (-1)(-1)?
Is (-1)1/2 defined?
What is it?
What is (-1)(-1)(-1)?
Is (-1)1/3 defined? What is it?
What (if they exist) are:
(-8)2/3
(-8)-(4/3)
(-4)-2/3
What is 0x0?
Is 01/2 defined?
What is it?
Is 0-1/2 defined?
What is 0x0x0?
Is 01/3 defined? What is it?
Is 0-1/3 defined?
(-4)2/3
SUMMARY:
Positive numbers can be raised to any power.
Zero can be raised to any positive power but
not to the zero or any negative power. The answer
is always 0.
Negative numbers can be raised to integer
powers and to fractional powers with odd
denominator (in lowest terms). The value is
positive if the numerator is even, otherwise
negative.
Which of the following exist?
-1-1
3-5/3
(-3)3/7
0-2
(-3/5)0 (-2)-2
(5/3)5/3 (-2/3)-2/3
01/2
(-2)1/2
NOTE: Most calculators will give an error
message if you try to evaluate any power of a
negative number. If you need the answer:
*check if it exists
*take the absolute value
*find the power
*correct the sign if necessary.
EXAMPLE: (-3)3
The exponent is an integer so the power exists.
Compute 33 = 27. The exponent is odd, so
(-3)3 = -27.
EXAMPLE: (-5/2)5/2
The denominator is even so the power does not
exist in the real numbers.
EXAMPLE: (-5/2)2/5
The denominator is even so the power exists.
(5/2)2/5  1.4427
The numerator is even so the power is positive.
(-5/2)2/5 1.4427
The TI83 actually gets this right (sometimes)
Compare (-3)^0.252
[63/250]
ERR: NONREAL ANSWER
and
(-3)^0.248
[31/125]
-1.313185477
It will compute (-3)^(3/7) correctly, but if you
evaluate 3/7 first, store it, and then try to raise
(-3) to that power it can no longer recognize it as a
rational number and it fails.
ROOTS
The nth root of a number is the same as the
1/nth power.