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

Question

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

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

Bunuel · Accepted Answer

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.

Carcass · Answer

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

PareshGmat · Answer

5^12 can be written as 25^6
When n=6, the equation becomes equal, so 7 should be the answer

Answer = B

JeffTargetTestPrep · Answer

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.

BrentGMATPrepNow · Answer

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

MHIKER · Answer

25^n > 5^{12}

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

=2n>12

=n>6

The smallest value n can take is 7

The answer is B.

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=W9nFosP6NMk

EMPOWERgmatRichC · Answer

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:  B

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

GMATinsight · Answer

Answer: Option B

Step-by-Step Video solution by  GMATinsight

https://www.youtube.com/watch?v=yaY6yR1bQXE

DanTheGMATMan · Answer

Get those like bases and then you're done:
 https://www.youtube.com/watch?v=JrSaUQ9yeJc

totaltestprepNick · Answer

B https://www.youtube.com/watch?v=93W6hKpn8rY Nick Slavkovich, GMAT/GRE tutor with 20+ years of experience tutoring@totaltestprep.net

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 ?

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