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?

CalculatorUser May 24, 2019

#2**+1 **

I am not saying that your answer is wrong CalculatorUser BUT did you consider the possibility that n could be negative?

Melody May 25, 2019

#3**+3 **

n^{72} < 5^{96} < (n+1)^{72}

n < 5^{96/72} < n+1

n < 5^{4/3} < n+1

n < 8.55 < n+1

n = 8

Alan May 25, 2019

#4**+1 **

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

#5**+1 **

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

CalculatorUser May 25, 2019

#6**+1 **

I suggest you think very hard about THIS answer/question

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