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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, 09:37
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Difficulty:

5% (low)

Question Stats:

88% (00:37) correct 12% (00:49) wrong based on 1995 sessions

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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
Math Expert
Joined: 02 Sep 2009
Posts: 52905
Re: What is the smallest integer n for which 25^n > 5^12 ?  [#permalink]

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10 Dec 2012, 09:41
3
1
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, 12:40
3
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, 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]

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Updated on: 10 Apr 2013, 22:20
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, 00:06.
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]

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07 Apr 2013, 06:25
1
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, 00:47
1
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, 09:53
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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09 Jun 2016, 13:10
2
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, 21:46
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.

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Posts: 147
Re: What is the smallest integer n for which 25^n > 5^12 ?  [#permalink]

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25 Mar 2017, 05:19
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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06 Mar 2018, 13:12
Top Contributor
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.

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

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20 Mar 2018, 14:45
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

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

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21 Mar 2018, 04:24
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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19 Oct 2018, 09:56
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.
Re: What is the smallest integer n for which 25^n > 5^12 ?   [#permalink] 19 Oct 2018, 09:56
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