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What is the smallest integer n for which 25^n > 5^12 ?

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What is the smallest integer n for which 25^n > 5^12 ? [#permalink] New post 10 Dec 2012, 09:37
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What is the smallest integer n for which 25^n > 5^12 ?

(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
[Reveal] Spoiler: OA
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Re: What is the smallest integer n for which 25^n > 5^12 ? [#permalink] New post 10 Dec 2012, 09:41
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Walkabout wrote:
What is the smallest integer n for which 25^n > 5^12 ?

(A) 6
(B) 7
(C) 8
(D) 9
(E) 10


Work with the common base: 25^n=(5^2)^n=5^{2n}.

Thus we have that 5^{2n}>5^{12} --> 2n>12 --> n>6 --> n_{min}=7.

Answer: B.
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Re: What is the smallest integer n for which 25^n > 5^12 ? [#permalink] New post 10 Dec 2012, 12:40
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Re: What is the smallest integer n for which 25^n > 5^12 ? [#permalink] New post 06 Apr 2013, 22:48
When i worked this problem I did the following:

5^12 -> 5*5^11 but why is 5*5^11 not equal 25^11?

I am stuck on this part, and im sure it is simply to do with order of operations but i cant wrap my mind around it.
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Re: What is the smallest integer n for which 25^n > 5^12 ? [#permalink] New post 07 Apr 2013, 00:06
specialxknc wrote:
When i worked this problem I did the following:

5^12 -> 5*5^11 but why is 5*5^11 not equal 25^11?

I am stuck on this part, and im sure it is simply to do with order of operations but i cant wrap my mind around it.


5*5^11 = 5^1 * 5^11 = 5*(1+11)

25^11 = (5^2)^11 = 5^(2*11) = 5^22

Rules for exponents are as follows:

(1) x^m * x^n = x^(m+n)
(2) x^m / x^n = x^(m-n)
(3) x^(-m) = 1/x^m
(4) x^0 =1
(5) (x^m)^n = x^(mn)
(6) x^m * y^m = (xy)^m

Last edited by doe007 on 10 Apr 2013, 22:20, edited 4 times in total.
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Re: What is the smallest integer n for which 25^n > 5^12 ? [#permalink] New post 07 Apr 2013, 06:25
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specialxknc wrote:
When i worked this problem I did the following:

5^12 -> 5*5^11 but why is 5*5^11 not equal 25^11?

I am stuck on this part, and im sure it is simply to do with order of operations but i cant wrap my mind around it.


Because you dropped parenthesis. 5*5^11 is ambiguous, so on one hand it could equal (5*5)^11, which is 25^11. But on the other hand it might mean 5*(5^11), which is 5^12.

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Re: What is the smallest integer n for which 25^n > 5^12 ? [#permalink] New post 27 Feb 2014, 00:47
5^12 can be written as 25^6
When n=6, the equation becomes equal, so 7 should be the answer

Answer = B
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Re: What is the smallest integer n for which 25^n > 5^12 ?   [#permalink] 27 Feb 2014, 00:47
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