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# How many zeros are in the expansion of ?

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How many zeros are in the expansion of \(999,\!999,\!999,\!998^2\)?

Mar 31, 2020
edited by Guest  Mar 31, 2020
edited by nchang  Mar 31, 2020

#1
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Whats the number?

Mar 31, 2020
#3
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Cal why are u answering homework??

Mar 31, 2020
#4
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that has a 9 on the end, not an 8

Mar 31, 2020
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Hello nchang! I like your name, it probably reveals your last name in real life! >:D

Know:

A number only has a 0 in it if when factored, has 2 * 5 in it.

Solve:

When you factor 999,999,999,9982count how many (2 * 5)s it has.

Mar 31, 2020
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Mar 31, 2020
#7
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Aren't those for terminating zeros? I don't think that is what the question is asking though.

Mar 31, 2020
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Oh shoot, you are right. Hmmm. Perhaps you can simplify 999,999,999,9982 into multiple, smaller exponents. I can't think of anything else.

AnExtremelyLongName  Mar 31, 2020
#9
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maybe (1000000000000-2)^2?

Mar 31, 2020
#10
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Good idea!

_________

So by the (a - b)2 = (a + b)(a - b), can we work this further? I am struggling as much as you right now.

AnExtremelyLongName  Mar 31, 2020
#13
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1,000,000,000,002 * 999,999,999,996

How will we know how many zeros that has?

AnExtremelyLongName  Mar 31, 2020
#11
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maybe something like 1000000000000^2-1000000000000*2*2+4?

Mar 31, 2020
#12
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Wait a sec didnt you say its 999,999,999? If it is that then it should be (100,000,000-1)^2 right?

Mar 31, 2020
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nchang  Mar 31, 2020
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But you said that it ended with a 9 a few minutes ago.....

AltShaka  Mar 31, 2020
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But you said... wait now I'm super confused and I lost my train of thought

AnExtremelyLongName  Mar 31, 2020
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I never said that!

nchang  Mar 31, 2020
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It lit says above.... Oh sorry the end is an 9 not an 8

AltShaka  Mar 31, 2020
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I am just wondering how to do this without a calculator.

Mar 31, 2020
#20
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Here is hunch:

An expression like x * y keeps the number of zeroes as long as x and y do not contain (2 * 5) or any factor of ten like (25 * 4) etc.

Ex: one zero (10 * 11 = 110) four zeroes (100 * 100 = 10,000)

- In the current problem, the numbers obviously do not have any factor of ten. So:

1,000,000,000,002 * 999,999,999,996​ we count the zeroes, getting an answer of \(\boxed{11}\)?

Mar 31, 2020
edited by AnExtremelyLongName  Mar 31, 2020
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That is so clever! Thank you so much, I understand now.

Mar 31, 2020
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Agree  !!!!

I'm going to have to look at  AELN's  method.....very  exquisite    !!!!

CPhill  Mar 31, 2020
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Note  that  we  can  write

(1,000,000,000 ,000  - 2)^2

By  binomial expansion we have

( 1, 000,000,000,000)^2  -  4  ( 1,000,000,000,000)  +  4  =

(10^12)^2  -  4 ( 10^12)  +  4  =

(10)^24  - 4 (10^12)  + 4  =

1, 000,000,000,000,000,000,000,004

-                              4,000,000,000,000

_________________________________

999,999,999,996,000,000,000,004

11  zeros

Just as AELN  found....good job,  AELN    !!!!!

Mar 31, 2020
#24
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I like your method better! It insures 100% accuracy!

AnExtremelyLongName  Mar 31, 2020
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That is kinda like my idea! Thank you CPill! (btw, who is Allen)?

Mar 31, 2020
#26
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My abbreviation lol

AnExtremelyLongName  Mar 31, 2020
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AELN  =  shorthand  for AnExtremelyLong Name.....LOL!!!!

CPhill  Mar 31, 2020
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LOL! :)

Mar 31, 2020