how many zeros

[nanase]

Hello I am working on this question, can you please tell me if what I have done is correct? there will be 26 zeros.
I have no answer key for this question

 

[The Highlander]

Hello I am working on this question, can you please tell me if what I have done is correct? there will be 26 zeros.
I have no answer key for this question
View attachment 37939

You haven't finished this, @nanase, so we can only tell you that what you have done so far is correct. ("there will be 26 zeros." is not the correct final answer.)

The last (complete) line you did could be simplified to: 101⁵ × 5¹⁷ × 10²⁶ but you can't just then say that will have 26 trailing zeroes!

Think about what you would actually get if you did that multiplication.

Hint: I would suggest you now convert the 101⁵ first to (1.01 × 100)⁵ and thence to 1.01 × 10^? and the 5¹⁷ first to (0.5 × 10)¹⁷ and thence to 5 × 10^?.

Once you have done that, you should then be able to figure out what the (completed) multiplication would return as a (single) number multiplied by 10ⁿ and that should allow you to figure out how many trailing zeroes are involved if the multiplication by 10ⁿ is done.

Hope that helps.

 

[Dr.Peterson]

Hello I am working on this question, can you please tell me if what I have done is correct? there will be 26 zeros.
I have no answer key for this question
View attachment 37939

I think you're correct, though I would have written the details differently (focusing on prime factors). You've shown that the number is equal to 10^26 times a number with no factors of 10 (because it has no factors of 2).

 

[The Highlander]

Correction:

The paragraphs:-

"Hint: I would suggest you now convert the 101⁵ first to (1.01 × 100)⁵ and thence to 1.01 × 10^? and the 5¹⁷ first to (0.5 × 10)¹⁷ and thence to 5 × 10^?.​

​

Once you have done that, you should then be able to figure out what the (completed) multiplication would return as a (single) number multiplied by 10ⁿ and that should allow you to figure out how many trailing zeroes are involved if the multiplication by 10ⁿ is done."​

Should have read:-

"Hint: I would suggest you now convert the 101⁵ first to (1.01 × 100)⁵ and thence to 1.01⁵ × 10^? and the 5¹⁷ first to (0.5 × 10)¹⁷ and thence to 0.5¹⁷ × 10¹⁷.​

​

Once you have done that, you should then be able to figure out what the (completed) multiplication would return as a (single) number multiplied by 10ⁿ and that should allow you to figure out how many trailing zeroes are involved if the multiplication by 10ⁿ is done."​

 

[Dr.Peterson]

("there will be 26 zeros." is not the correct final answer.)

The last (complete) line you did could be simplified to: 101⁵ × 5¹⁷ × 10²⁶ but you can't just then say that will have 26 trailing zeroes!

I guess I need to directly challenge you. WHY can't you conclude there are 26 trailing zeros? And HOW does introducing decimals help? I wonder if you are thinking about a different question.

 

[The Highlander]

I think you're correct, though I would have written the details differently (focusing on prime factors). You've shown that the number is equal to 10^26 times a number with no factors of 10 (because it has no factors of 2).

Indeed and that's a much quicker and much more elegant method than what I suggested!
(I also said that 26 zeroes was wrong before I worked it out myself; my bad! )

Following my suggestion (which I now realize wasn't very clever) the final line in the OP could be rearranged to:-

\(\displaystyle \quad 1.01^5\times 10^{10}\times 0.5^{17}\times10^{17}\times 10^{26}\\\,\\=1.01^5\times 0.5^{17}\times 10^{53}\\\,\\=8.018570328521728515625\times 10^{-6}\times 10^{53}\\\,\\=8.018570328521728515625\times 10^{47}\\\,\\=8,018,570,328,521,728,515,625\times 10^{-21}\times 10^{47}\\\,\\=\underline{\underline{8,018,570,328,521,728,515,625\times 10^{26}}}\)

But you need a BIG Calculator to do that and once you have such a calculator you can just as easily enter:

\(\displaystyle 101^5\times 5^{17}\times 10^{26}\) (without any further manipulation)

and get: \(\displaystyle \underline{\underline{801,857,032,852,172,851,562,500,000,000,000,000,000,000,000,000}}\)
(which has 26 trailing zeroes ).

This appeared whilst I was composing the above...

I guess I need to directly challenge you. WHY can't you conclude there are 26 trailing zeros? And HOW does introducing decimals help? I wonder if you are thinking about a different question.

No, I wasn't thinking about a different question, I just wasn't thinking very well at all! (I'm tired. )

I simply didn't see your solution or even consider the factors. It just struck me that the mantissa (of the final product) would have multiple decimal places (and quite possibly an index that would affect the 10²⁶ that was present, hence leading to a different number of trailing zeroes); clearly, I was wrong!

I had also suspected that the numbers given would lead to a product that was a nice round number (with fewer than 8 decimal places) but a few quick presses on my 10 digit device (after my 1st post) soon disabused me of that notion and I had to use an online calculator (set to 60 d.p.) to confirm that my (hugely overly laborious) method worked.

So your "challenge" goes completely unopposed.