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

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

$$25^n > 5^{12}$$

$$=5^{2n}>5^{12}$$

$$=2n>12$$

$$=n>6$$

The smallest value n can take is 7

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

We’re asked for the SMALLEST possible INTEGER value for N that will make 25^N > 5^12. While the answers to this question are numbers (so we could TEST THE ANSWERS), this prompt is based around some standard Exponent rules, so approaching it with Arithmetic should be fairly quick.

The “base” of an exponent calculation can sometimes be ‘re-written’ (if the base has any factors that are greater than 1, then you can rewrite the calculation while keeping its value the same). Here, 25 can be re-written as 5^2, so the “left side” of the inequality can be re-written as (5^2)^N.

When ‘raising a power to a power’, you multiply the Exponents, meaning that we now have:

5^(2N) > 5^12

Thus, we need 2N > 12…. N > 6. The smallest answer that fits that description is…

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