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

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10 Dec 2012, 10:41
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Expert's post
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 $$5^{2n}>5^{12}$$ --> $$2n>12$$ --> $$n>6$$ --> $$n_{min}=7$$.

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

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10 Dec 2012, 13:40
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Here we have to be careful

because we know that n is > 6. So be on the lookout to not choose 6 as answer$$BUT 7$$

During the test such error could be common
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Re: What is the smallest integer n for which 25^n > 5^12 ? [#permalink]

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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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07 Apr 2013, 01:06
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, 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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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.

Kind Regards,

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

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27 Feb 2014, 01:47
5^12 can be written as 25^6
When n=6, the equation becomes equal, so 7 should be the answer

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

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11 Sep 2014, 10:53
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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25 Dec 2015, 21:34
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Re: What is the smallest integer n for which 25^n > 5^12 ? [#permalink]

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09 Jun 2016, 14:10
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.

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

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09 Jun 2016, 22:46
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.

Re: What is the smallest integer n for which 25^n > 5^12 ?   [#permalink] 09 Jun 2016, 22:46
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