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Exercise 1.2.17: Let us suppose we know x¹/" exists for every x>0 and every nẸN (see Exercise 1.2.11 above). For integers p and q>0 where P/a is in lowest terms, define xP/9 := (x¹/9)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 (x¹/4)= (x¹/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 y". c) Suppose x>1 and r,s are rational where r r. Hint: Writer and s with the same denominator.

Exercise 1.2.17: Let us suppose we know x¹/" exists for every x>0 and every nẸN (see Exercise 1.2.11 above). For integers p and q>0 where P/a is in lowest terms, define xP/9 := (x¹/9)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 (x¹/4)= (x¹/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 y". c) Suppose x>1 and r,s are rational where r r. Hint: Writer and s with the same denominator.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 1.2.17: Let us suppose we know x¹/n exists for every x>0 and every nEN (see Exercise 1.2.11 above). For integers p and q> 0 where P/q is in lowest terms, define xP/4 = (x¹/9). 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 (x¹/9)P = (x¹/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" <y. Then suppose r <0 and show: x<y if and only if x' >y". c) Suppose x>1 and r,s are rational where r<s. Show x" <x. If0<x< 1 and r<s, show that x'>r Hint: Writer and s with the same denominator. d) (Challenging)* For an irrational z € R\Q and x > 1 define x := sup{x' :r<z,r € Q}, for x = 1 define 1² = 1, and for 0<x< 1 define x := inf{x": r ≤z,r € Q}. Prove the two assertions of part b) for all real z
Transcribed Image Text:Exercise 1.2.17: Let us suppose we know x¹/n exists for every x>0 and every nEN (see Exercise 1.2.11 above). For integers p and q> 0 where P/q is in lowest terms, define xP/4 = (x¹/9). 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 (x¹/9)P = (x¹/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" <y. Then suppose r <0 and show: x<y if and only if x' >y". c) Suppose x>1 and r,s are rational where r<s. Show x" <x. If0<x< 1 and r<s, show that x'>r Hint: Writer and s with the same denominator. d) (Challenging)* For an irrational z € R\Q and x > 1 define x := sup{x' :r<z,r € Q}, for x = 1 define 1² = 1, and for 0<x< 1 define x := inf{x": r ≤z,r € Q}. Prove the two assertions of part b) for all real z
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Step 1

As per the Bartleby guidelines, we can solve only first three sub parts. Please re post the question and request for the fourth sub part.

Given Information:

For any x>0, x1n exists.

For any p∈ℤ and q>0, then xpq=x1qp, where pq in lowest form.

To prove:

a) Power is well defined even the fraction is not in lowest form.

b) x,y>0 and x,y∈ℤ, r∈ℚ

Case i) r>0, x<y⇔xr<yr

Case ii) r<0, x<y⇔xr>yr

c) r,s∈ℚ and r<s with same denominator.

Case i) x>1, xr<xs

Case ii) 0<x<1, xr>xs

Exponent laws:

i) amn=amn

ii) amn=am1n

iii) abm=ambm

iv) a-m=1am

Inequality rule:

i) ad<bc⇔ab<cd

ii) ad>bc⇔ab>cd

iii) a<b⇔-a>-b

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