Basic Analysis I — Real Analysis · Problem 1.2.17

Real Numbers — Jiri Lebl

Mathematics — Real Analysis · rational-exponents

Problem

Let us suppose we know

x^{1/n}

exists for every

x > 0

and every

n \in \mathbb{N}

(see Exercise 1.2.11 above). For integers

p

and

q > 0

where

p/q

is in lowest terms, define

x^{p/q} := (x^{1/q})^p

. a) Show that the power is well-defined even if the fraction is not in lowest terms: If

p/q = m/k

where

m

and

k > 0

are integers, then

(x^{1/q})^p = (x^{1/k})^m

. b) Let

x

and

y

be two positive numbers and

r

a rational number. Assuming

r > 0

, show

x < y

if and only if

x^r < y^r

. Then suppose

r < 0

and show:

x < y

if and only if

x^r > y^r

. c) Suppose

x > 1

and

r, s

are rational where

r < s

. Show

x^r < x^s

. If

0 < x < 1

and

r < s

, show that

x^r > x^s

. Hint: Write

r

and

s

with the same denominator. d) (Challenging) For an irrational

z \in \mathbb{R} \setminus \mathbb{Q}

and

x > 1

define

x^z := \sup\{x^r : r \leq z, r \in \mathbb{Q}\}

, for

x = 1

define

1^z = 1

, and for

0 < x < 1

define

x^z := \inf\{x^r : r \leq z, r \in \mathbb{Q}\}

. Prove the two assertions of part b) for all real

z