Solutions To Mathematics Textbooks/Basic Mathematics/Chapter 1

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Contents

Chapter 1Edit

Section 2Edit

1Edit

 

  by associativity
  by commutativity
  by commutativity
  by associativity

14Edit

 

 

 

15Edit

 
 
  multiply both sides by -1.

Section 3Edit

30, 31, 32, 33Edit

Although these can be done by hand the exercises suggest that you derive a general formula for finding the final population figures. The following one will suffice for the stated problems, where   is the initial population,   is the scaling factor (doubles, triples, etc.),   and   is the inital year and end year respectively, and   is the number of years taken for the population to go up by  :


 


For example, 32a:


 


Section 4Edit

8-15Edit

Write out the powers of 2 not exceeding the largest number in the problem set. Systematically check to see if each power divides the given number, starting with the largest.

16-23Edit

Write out the powers of 3 not exceeding the largest number in the problem set. Systematically check to see if each power divides the given number, starting with the largest.

24Edit

a)Edit

Examining the definitions first, if   and   then   and  .


  means that  .


 

 

 


Let  


 

b)Edit

Examine the definitions in part a) again. We have   and  .


Solving for   and  :


  and  


 

 

 


Let  .


 

 

 .

Section 5Edit

7Edit

a)Edit

120, 720, 5040 and 40320.

c)Edit

 


 ,


 ,


 


d)Edit

 


 


Mutliply both sides of the equation by   and  , cancelling unecessary factors to 1. We do this in order to achieve the denominator  :


 


Use the fact that   and   to achieve:


 


 


 


 


 


Hence,


 


Section 6Edit

1Edit

a)Edit

 


 

 

 


 

2Edit

a)Edit

 


 


 


 

b)Edit

 

Recall from the chapter that if  , then  .