Question 75572
<pre><font size = 5><b>You didn't give any problems,
so I'll make some up. I hope they are
like the ones you were asked to solve.

All rational numbers expressed
as decimals will either terminate
or repeat a block of digits
forever.  That's because in long
division all the remainders will
always be less than the divisor,
so sooner or later the remainder
will either be 0 or will be the
repeat of a remainder obtained
earlier.

write the decimal expansions for these 
rational numbers. tell whether the 
expansions are terminating or repeating. 

#1.   9/32

Use long division:

  <u>  .28125</u> 
32)9.00000
   <u>6 4</u>
   2 60
   <u>2 56</u>
      40
      <u>32</u>
       80
       <u>64</u>
       160
       <u>160</u>
         0

Terminating because the division 
eventually has 0 remainder

#2. 15/22

Again use long division:

  <u>   .6818</u> 
22)15.0000 
   <u>13 2</u>
    <font color = "red">1 8</font>0
    <u>1 76</u>
       40
       <u>22</u>
       <font color = "red">18</font> 

This is a repeating decimal because the same 
non-zero remainder 18 occurred twice in the 
long division. These two 18's are indicated in 
red above.


--------------------------------------------

Find the rational number equivalents for these 
decimal expansions. tell whether they are 
rational or irrational

#3.   .185185185185···

Let N = .<font color = "red">185</font><font color = "blue">185</font><font color = "red">185</font><font color = "blue">185</font>···
There are 3 digits in the repeating block
"185", so multiply by 10<sup>3</sup> or
1000

1000N = 185.185185185···

Now place the first equation underneath
and subtract the two equations

1000N = 185.185185185···
    N =    .185185185185···
 999N = 185.000000000
 999N = 185
    N = 185/999
that reduces to
    N = 5/27

This is RATIOnal because it is the
RATIO of two integers

#2.  2.9373737837···

Let N = 2.9<font color = "red">37</font><font color = "blue">37</font><font color = "red">37</font><font color = "blue">37</font>··· 

There are 2 digits in the repeating block
"37", so multiply by 10<sup>2</sup> or
100

100N = 293.737373737···

Now place the first equation underneath
and subtract the two equations

100N = 293.737373737···
   N =   2.93737373737···
 99N = 290.800000000···
 99N = 290.8
Clear of decimals by multiplying thru
by 10

990N = 2908
   N = 2908/990
that reduces to
    N = 1454/495

This is RATIOnal because it is the
RATIO of two integers
  
#3.  .<font color = "red">7</font><font color = "blue">2</font><font color = "red">77</font><font color = "blue">22</font><font color = "red">777</font><font color = "blue">222</font><font color = "red">7777</font><font color = "blue">2222</font>···

This is IRRATIONAL because  
there is no block of repeating
digits.

--------------------------------------------

simplify the following 
radicals. 

#4.  {{{sqrt(252) = sqrt(36*7) = sqrt(36)sqrt(7) = 6sqrt(7)}}}

#5. {{{sqrt(8.67) = sqrt(867/100) = sqrt(867)/sqrt(100) = sqrt(289*3)/10 = (sqrt(289)sqrt(3))/10 = (17sqrt(3))/10}}}

Edwin</pre>