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I need help solving this, could someone please solve and explain?

 Oct 15, 2014

Best Answer 

 #2
avatar+23254 
+5

Written as unreduced radicals and arranged in order, the problem is:

1/(√16 + √14) + 1/(√14 + √12)+1/(√12 + √10)+ 1/((√10 + √8) * 1/(√8 + √6) + 1/(√6 + √14)

Now multiply each fraction by (its conjugate divided by its conjugate):

1/(√16 + √14)·(√16 - √14)/(√16 - √14) =  (√16 - √14)/(16 - 14) = (√16 - √14)/2

1/(√14 + √12)·(√14 - √12)/(√14 - √12) =  (√14 - √12)/(14 - 12) = (√14 - √12)/2

1/(√12 + √10)·(√12 - √10)/(√12 - √10) =  (√12 - √10)/(12 - 10) = (√12 - √10)/2

...

1/(√6 + √4)·(√6 - √4)/(√6 - √4) =  (√6 - √4)/(6 - 4) = (√6 - √4)/2

Note that all the denominators are 2, so the numerators can be added.

Now note that when you add all the numerators together, all the intermediate terms cancel and the only terms left in the numerator is the first and the last:  √16 - √4  = 4 - 2  = 2

So this sumes to  2/2  = 1  <-----  A quite surprising result! And, a super neat problem!

 Oct 15, 2014
 #1
avatar
0

sorry the image wouldn't get in.

Problem 32 on the individual test here.

 Oct 15, 2014
 #2
avatar+23254 
+5
Best Answer

Written as unreduced radicals and arranged in order, the problem is:

1/(√16 + √14) + 1/(√14 + √12)+1/(√12 + √10)+ 1/((√10 + √8) * 1/(√8 + √6) + 1/(√6 + √14)

Now multiply each fraction by (its conjugate divided by its conjugate):

1/(√16 + √14)·(√16 - √14)/(√16 - √14) =  (√16 - √14)/(16 - 14) = (√16 - √14)/2

1/(√14 + √12)·(√14 - √12)/(√14 - √12) =  (√14 - √12)/(14 - 12) = (√14 - √12)/2

1/(√12 + √10)·(√12 - √10)/(√12 - √10) =  (√12 - √10)/(12 - 10) = (√12 - √10)/2

...

1/(√6 + √4)·(√6 - √4)/(√6 - √4) =  (√6 - √4)/(6 - 4) = (√6 - √4)/2

Note that all the denominators are 2, so the numerators can be added.

Now note that when you add all the numerators together, all the intermediate terms cancel and the only terms left in the numerator is the first and the last:  √16 - √4  = 4 - 2  = 2

So this sumes to  2/2  = 1  <-----  A quite surprising result! And, a super neat problem!

geno3141 Oct 15, 2014
 #3
avatar+130511 
0

I like that one, too!!

 

 Oct 15, 2014
 #4
avatar+118723 
0

I can't find problem 32 on that test.

It's all a mystery to me.    

 Oct 15, 2014

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