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200^200 * 40^40

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200^200 * 40^40 [#permalink] New post 22 Mar 2018, 12:28
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Question Stats:

70% (02:31) correct 30% (03:02) wrong based on 10 sessions
\(\frac{200^{200} 40^{40}}{20^{20}400^{400}}\)

A. \(\frac{1}{2^{920}5^{380}}\)

B. \(\frac{1}{2^{240}5^{240}}\)

C. \(\frac{1}{2^{180}5^{180}}\)

D. \(1\)

E. \(2^{180}5^{180}\)
[Reveal] Spoiler: OA

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Re: 200^200 * 40^40 [#permalink] New post 22 Mar 2018, 18:58
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It's always good when practicing to consider guessing strategies. With this problem, we see 3 answer choices that are very small, one that is 1, and another that is quite large. This disparity means that guessing is a possibility, at least partially. Notice that the largest base, 400, also has the largest exponent, 400. This number will certainly dominate all the others, and since it's in the denominator, we at least know that the answer will be less than 1, eliminating D and E.

Let's actually solve this thing. Anytime you see numbers with huge exponents, it's a good thought to break them down to their primes so we can cancel as quickly and easily as possible. And anytime you see numbers ending with zeroes, you can think of each zero as a factor of 10, but more usefully as factors of 2 and 5. So each of these bases (200, 40, 20, and 400) can be broken down into various combinations of 2s and 5s. We could thus break down each into its 2s and 5s and then apply the exponent to each of them and then see how many cancel out. However, notice that each answer choice shows different exponents for each of the 2s. This tells us that we don't even need the 5s. So let's determine how many 2s are factors in each term and leave it at that.

For 200^200, we know the base ends in two 0s, each of which can be thought of as a 5 and a 2. So that's two 2s so far. Add one more 2 since the whole thing starts with a 2, so we know that 200 has three 2s in its prime factorization. If we rewrite it, it would look like this: (2x2x2x5x5)^200. Ignoring the 5s, we get 2^600.

Similarly, the 40^40 can quickly be broken down into (2x2x2x5)^40, giving us 2^120.

In the denominator, we have 20^20, breaking down into (2x2x5)^20 and giving us 2^40, and 400^400, breaking down into (2x2x2x2x5x5)^400, giving us 2^1600.

To recap, ignoring the 5s, we have (2^600x2^120)/(2^40x2^1600). I'd cancel the exponent of 600 with the exponent of 1600, leaving a 2^1000 in the denominator, and cancel the exponent of 120 with the exponent of 40, leaving 2^80 in the numerator. At this point we have 2^80/2^1000, which leaves us with 1/2^920. Note that only answer choice A has this in it, so the answer must be A, and we never had to deal with the 5s.
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Re: 200^200 * 40^40   [#permalink] 22 Mar 2018, 18:58
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