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# Which is greater? 54^200 or 15^300

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GMAT Club Legend
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Which is greater? 54^200 or 15^300 [#permalink]  24 Mar 2004, 11:03
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Which is greater? 54^200 or 15^300?
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Paul

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if we take both numbers under sqrt100 then it becomes 54^2 vs. 15^3 which is 54^2=2916 15^3=3375 so the second one is bigger
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BG wrote:
if we take both numbers under sqrt100 then it becomes 54^2 vs. 15^3 which is 54^2=2916 15^3=3375 so the second one is bigger

it is actually the simplest way to solve this problem
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Paul

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Re: PS: exponents [#permalink]  24 Mar 2004, 11:27
Paul wrote:
Which is greater? 54^200 or 15^300?

Can i do it this way?

1. 54^200 = 3^600 * 2^200 = 3^300 * 3^300 * 2^200

2. 15^300 = 3^300 * 5^300 = 3^300 * 5^300

Since 300log5 > (300log3 + 200log2)

implies 2 > 1.
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How else can you solve this without knowing that 300log5 > (300log3 + 200log2)?
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ndidi204 wrote:
How else can you solve this without knowing that 300log5 > (300log3 + 200log2)?

I think this can be the alternative:

We need to check which one is greater:

1) 5^300

or

2) (3^300 * 2^200)

Lets divide the two

(1)/(2)

= (5^300)/(3^300 * 2^200)
= (5^300)(2^100)/(3^300 * 2^300)
= (2^100)/(1.2^300)
= (2^100)/(1.732^100)
= (2/1.732)^100 > 1

Please let me know if we have another solution.

Thanx
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The first term = 54^200 = [X^(3/2)]^200 = X^300

The second term is = 15^300

So we need to find which is greater X or 15?

We know that X^(3/2) = 54 = (3^3 * 2)

Hence X^3 = 3^6 * 4

=> X = 3^2 * 4^(1/3)

So X is < 9 * 1.6

=> X is < 14.4

So the second term is greater.
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