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What is the smallest integer a for which 27^a>3^24?

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Bunuel
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What is the smallest integer a for which 27^a>3^24? [#permalink] New post   29 Feb 2016, 11:06
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What is the smallest integer a for which 27^a > 3^24?

A. 7
B. 8
C. 9
D. 10
E. 12
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C
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Re: What is the smallest integer a for which 27^a>3^24? [#permalink] New post   29 Feb 2016, 11:43
Bunuel wrote:
What is the smallest integer a for which 27^a > 3^24?

A. 7
B. 8
C. 9
D. 10
E. 12



27= 3^3
hence, (3^3)^a
Option C would make it 3^27 > 3^24
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Re: What is the smallest integer a for which 27^a>3^24? [#permalink] New post   03 Mar 2016, 03:12
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Bunuel wrote:
What is the smallest integer a for which 27^a > 3^24?

A. 7
B. 8
C. 9
D. 10
E. 12


27^a > 3^24
Converting into the same bases:
27^a > 27^8
Therefore for the equation to hold true, a > 8 or a = 9
Option C
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What is the smallest integer a for which 27^a>3^24? [#permalink] New post   03 Mar 2016, 18:10
Simply and how else is there to do so then:
\(3^{3a} > 3^{24}\)
\(3a > 24\)
\(a > 8\)

Therefore, the smallest integer is 9.

C.
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Re: What is the smallest integer a for which 27^a>3^24? [#permalink] New post   09 Mar 2016, 23:43
Nice Question..
Here just need to write 27 as 3^3 and we are able to see that a must be atleast 9
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Re: What is the smallest integer a for which 27^a>3^24? [#permalink] New post   21 Mar 2018, 03:21
Bunuel wrote:
What is the smallest integer a for which 27^a > 3^24?

A. 7
B. 8
C. 9
D. 10
E. 12


\(27^a > 3^{24}\)

or, \(3^{3a} > 3^{24}\)

when "a" = 9

\(3^{3*9} > 3^{24}\)

\(3^{27} > 3^{24}\)

Hence (B)
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Re: What is the smallest integer a for which 27^a>3^24? [#permalink] New post   22 Mar 2018, 15:44
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Bunuel wrote:
What is the smallest integer a for which 27^a > 3^24?

A. 7
B. 8
C. 9
D. 10
E. 12


Re-expressing 27 as 3^3, we have:

3^3a > 3^24

When bases are equal, we can deal with just the exponents, as follows:

3a > 24

a > 8

Answer: C
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Re: What is the smallest integer a for which 27^a>3^24? [#permalink] New post   26 Mar 2018, 10:52
Expert Reply

Solution:



Given:

    • a is an integer.

    • \(27^a > 3^{24}\).

Working out:

We need to find the smallest value of the integer “a”.

The given inequality is \(27^a >3^{24}\)

\(27^a\) can be written as \((3^3)^a\)

So, our inequality now becomes: \(3^3a > 3^{24}\)

    • Since the bases are same, we can equate the exponents.

    • Thus, \(3a> 24\)

    • Or, \(a>8\)

The smallest integer greater than 8 is 9, and hence \(a =9\)

Answer: Option C
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Re: What is the smallest integer a for which 27^a>3^24? [#permalink] New post   06 May 2020, 09:18
Bunuel wrote:
What is the smallest integer a for which 27^a > 3^24?

A. 7
B. 8
C. 9
D. 10
E. 12


Asked: What is the smallest integer a for which 27^a > 3^24?

\(27^a = 3^{3a} > 3^24\)
3a > 24
a > 8
\(a_{min} = 9\)

IMO C
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Re: What is the smallest integer a for which 27^a>3^24? [#permalink]
06 May 2020, 09:18
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