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# I am not sure if this is right. Exponential Inequalities.

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What integer n has the property that \(5^{96}\) is greater than \(n^{72}\) and \(5^{96}\) is less than \((n+1)^{72}\).

So I did

5^96>n^72

5^96<(n+1)^72

Then take 12th root

5^8>n^6

5^8<(n+1)^6

Then I take square root

5^4>n^3

5^4<(n+1)^3

Then I easily deduced that n must be 8.

Is my solution correct?

May 24, 2019

#1
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Well.....   there's one sure way to know......

5^96 > 8^72  ????  True

5^96 < (9)^72 ???? True

Looks like your solution is just fine, CU!!!

Very ingenious !!!!

May 24, 2019
edited by CPhill  May 24, 2019
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I am not saying that your answer is wrong CalculatorUser  BUT did you consider the possibility that n could be negative?

May 25, 2019
#3
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n72 < 596 < (n+1)72

n < 596/72 < n+1

n < 54/3 < n+1

n < 8.55 < n+1

n = 8

May 25, 2019
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I know that is right Alan and I think CalculatorUser does too, he wanted to do it without the calc I think.

BUT I just wanted him/her to think about wether there could be a negative answer.

I know the answer to this, I just want to be sure that CalculatorUser has thought about it too.

Melody  May 25, 2019
edited by Melody  May 25, 2019
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Yes thank you, Melody, I never thought that my answer could be negative. I am lucky that the problem only had one answer.

Next time in these problems, I will make sure there are no negative answers

May 25, 2019
#6
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You should not just accept what some answer written down somewhere is.

How do you know there is not a negative answer?  Answer: Someone told you so.

Well that is not you knowing, that is at best someone else knowing.

If you think about it purely as a logic exercise, you will work out why there cannot be a negative answer.

I took me a while to get my head around it so it might take you a while too but the reason is not very complex.

Melody  May 26, 2019