Math 140A, Fall 2010, Quiz, 10/15/10
Instructions
. Answer all questions. You may use without proof anything which was
proved in class. If you need to cite a theorem, do so either by name, if it has one, or by
brieFy stating what it says.
1. (10 points) Let
a, b
be positive real numbers,
p
and
q
nonzero positive integers.
Explain what it means for
b
1
/p
=
a
. Using this, show that
(
b
1
/p
)
1
/q
=
b
1
/pq
.
b
1
/p
=
a
if
a
is the unique positive number such that
a
p
=
b
.
Note that (
a
q
)
p
=
a
pq
since
p, q
are integers, for
a
a positive real number. If
a
=
b
1
/pq
,
then
b
=
a
pq
= (
a
q
)
p
. Thus
a
q
=
b
1
/p
, so
a
= (
b
1
/p
)
1
/q
.
2. (10 points) Show that if
x
∈
R
k
and
x
·
y
= 0 for all
y
∈
R
k
, then
x
=
0
.
Suppose
x
n
= 0, say
x
= (
x
1
, . . ., x
k
) with some
x
i
n
= 0. Then let
y
be the vector all
of whose components are zero except the
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 staff
 Math, positive real numbers, positive real number, Dedekind cut, unique positive number

Click to edit the document details