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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]

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

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New post 10 Dec 2012, 10:41
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Re: What is the smallest integer n for which 25^n > 5^12 ?  [#permalink]

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New post 10 Dec 2012, 13:40
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Re: What is the smallest integer n for which 25^n > 5^12 ?  [#permalink]

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New post 06 Apr 2013, 23:48
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]

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New post Updated on: 10 Apr 2013, 23:20
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) \(x^m * y^m\) = \((xy)^m\)

Originally posted by doe007 on 07 Apr 2013, 01:06.
Last edited by doe007 on 10 Apr 2013, 23:20, edited 4 times in total.
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Re: What is the smallest integer n for which 25^n > 5^12 ?  [#permalink]

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New post 07 Apr 2013, 07:25
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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]

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New post 27 Feb 2014, 01:47
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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]

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New post 11 Sep 2014, 10:53
Walkabout wrote:
What is the smallest integer n for which 25^n > 5^12 ?

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



25^n > 5^12

5^2n>5^12

2n>12

n has to be minimum 7
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Re: What is the smallest integer n for which 25^n > 5^12 ?  [#permalink]

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New post 09 Jun 2016, 14:10
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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


To solve, we want to get the bases the same. Thus we need to break 25^n into prime factors.

25^n = (5^2)^n = 5^(2n) (Remember that when we have a power to a power, we multiply the exponents.)

We can use the new value in the given inequality:

5^(2n)> 5^12

Since we have the same bases on either side of the inequality we can drop the bases and set up an equation involving just the exponents.

2n > 12

n > 6

Because n is greater than 6, the smallest integer that satisfies the inequality 25^n > 5^12 is 7.

The answer is B.
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Re: What is the smallest integer n for which 25^n > 5^12 ?  [#permalink]

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New post 09 Jun 2016, 22:46
Walkabout wrote:
What is the smallest integer n for which 25^n > 5^12 ?

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


\(25^n\) > \(5^12\)
\(5^2n\) > \(5^12\)

Since 5 is a positive number

2n> 12

n > 6

Smallest integer value for which n> 6: 7.

B is the answer
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Re: What is the smallest integer n for which 25^n > 5^12 ?  [#permalink]

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New post 25 Mar 2017, 06:19
be careful here, we are not looking for an answer that would give us >=0, but >=
5^2n>5^12
n is 7
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Re: What is the smallest integer n for which 25^n > 5^12 ?  [#permalink]

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New post 06 Mar 2018, 14:12
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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


We have: 25^n > 5^12

To rewrite this inequality with the SAME base, we'll replace 25 with .
When we do so, we get: ()^n > 5^12
Apply the Power of a Power law to get: 5^(2n) > 5^12

This means that it must be the case that 2n > 12
Divide both sides of the inequality by 2 to get: n > 6

What is the smallest integer n for which 25^n > 5^12 ?
We now know that n > 6
So, 7 is the smallest possible INTEGER value that satisfies this inequality.

Answer: B

Cheers,
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Re: What is the smallest integer n for which 25^n > 5^12 ?  [#permalink]

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New post 20 Mar 2018, 15:45
Walkabout wrote:
What is the smallest integer n for which 25^n > 5^12 ?

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


25^n > 5^12

5^2n > 5^12

Now we compare power

2n > 12...........n >6.......then n =7

Answer: 7
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Re: What is the smallest integer n for which 25^n > 5^12 ?  [#permalink]

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New post 21 Mar 2018, 05:24
Walkabout wrote:
What is the smallest integer n for which 25^n > 5^12 ?

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


\(25^n > 5^{12}\)

\(5^{2n} > 5^{12}\)

\(2n > 12\)

\(n > \frac{12}{2}\)

\(n > 6\)

Smallest value of "n" we require is "7"

Hence (B)
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Re: What is the smallest integer n for which 25^n > 5^12 ?  [#permalink]

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New post 19 Oct 2018, 10:56
I understand the solution but just a question.
Why is the following reasoning wrong?

25^n > 5^12
(5^2)^n > 5^12
5^(2^n) > 5^12
2^n > 12

And because 2^4=16, n=4.

Could somebody please explain the error in the calculations? Preferably, refer to the exponent rule that I am misunderstanding. Thank you in advance.
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Re: What is the smallest integer n for which 25^n > 5^12 ?  [#permalink]

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New post 20 Oct 2019, 01:53
What is the smallest integer n for which 25^n > 5^12 ?

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

Answer: C
25^n > 5^12
=> 5^2n > 5^12
=> 2n > 12
=> n > 6
=> n >=7 --> so the smallest integer value of n is 7
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Re: What is the smallest integer n for which 25^n > 5^12 ?   [#permalink] 20 Oct 2019, 01:53
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