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

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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We have: 25^n > 5^12

To rewrite this inequality with the SAME base, we'll replace 25 with 5².
When we do so, we get: (5²)^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,
Brent

Video solution from Quant Reasoning:
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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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5^12 can be written as 25^6
When n=6, the equation becomes equal, so 7 should be the answer

Answer = B

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

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

\(=2n>12\)

\(=n>6\)

The smallest value n can take is 7

The answer is B.
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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…

Final Answer:

GMAT Assassins aren’t born, they’re made,
Rich

Answer: Option B

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