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# M26-07

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Math Expert
Joined: 02 Sep 2009
Posts: 46129

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16 Sep 2014, 01:24
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Difficulty:

45% (medium)

Question Stats:

75% (01:17) correct 25% (01:38) wrong based on 28 sessions

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If $$x$$ is a positive integer then the value of $$\frac{22^{22x}-22^{2x}}{11^{11x}-11^x}$$ is closest to which of the following?

A. $$2^{11x}$$
B. $$11^{11x}$$
C. $$22^{11x}$$
D. $$2^{22x}*11^{11x}$$
E. $$2^{22x}*11^{22x}$$

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Joined: 02 Sep 2009
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16 Sep 2014, 01:24
Official Solution:

If $$x$$ is a positive integer then the value of $$\frac{22^{22x}-22^{2x}}{11^{11x}-11^x}$$ is closest to which of the following?

A. $$2^{11x}$$
B. $$11^{11x}$$
C. $$22^{11x}$$
D. $$2^{22x}*11^{11x}$$
E. $$2^{22x}*11^{22x}$$

Note that we need approximate value of the given expression. Now, $$22^{22x}$$ is much larger number than $$22^{2x}$$. Hence $$22^{22x}-22^{2x}$$ will be very close to $$22^{22x}$$ itself, basically $$22^{2x}$$ is negligible in this case. The same way $$11^{11x}-11^x$$ will be very close to $$11^{11x}$$ itself.

Thus $$\frac{22^{22x}-22^x}{11^{11x}-11^x} \approx \frac{22^{22x}}{11^{11x}}=\frac{2^{22x}*11^{22x}}{11^{11x}}=2^{22x}*11^{11x}$$.

You can check this algebraically as well: $$\frac{22^{22x}-22^{2x}}{11^{11x}-11^x}=\frac{22^{2x}(22^{20x}-1)}{11^x(11^{10x}-1)}$$. Again, -1, both in denominator and numerator is negligible value and we'll get the same expression as above: $$\frac{22^{2x}(22^{20x}-1)}{11^x(11^{10x}-1)} \approx \frac{22^{2x}*22^{20x}}{11^x*11^{10x}}=\frac{22^{22x}}{11^{11x}}=2^{22x}*11^{11x}$$.

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Joined: 14 Oct 2015
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28 Oct 2015, 08:06
I think this is a high-quality question and I agree with explanation. Doesn't the x go away in 11^11x?? I'm getting 2^22x*11^11
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Joined: 02 Sep 2009
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29 Oct 2015, 01:12
danjbon wrote:
I think this is a high-quality question and I agree with explanation. Doesn't the x go away in 11^11x?? I'm getting 2^22x*11^11

Check below:

$$\frac{22^{22x}}{11^{11x}}=\frac{(2*11)^{22x}}{11^{11x}}=\frac{2^{22x}*11^{22x}}{11^{11x}}=2^{22x}*11^{22x-11x}=2^{22x}*11^{11x}$$

Does it make sense?
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10 Mar 2018, 14:06
Great question

It very easy to waste a lot of time trying to solve this question algebraically (because it looks a little like you need to use difference of squares) if you miss the hint "closet to" and don't realise you're being asked to approximate, thus can take out 22^2X and 11^2X as factors and discard the +1s they produce when factoring out.

However, even if you don't notice that, you should realise that once you break down 22 to prime factors (22^22=2^22*11^22) you have powers of 2 up on the numerator and nothing to cancel that or decrease it on the denominator. In contrast, you do have powers of 11 on the denominator - so look for an answer choice where the exponent of the 2s is unchanged but the exponent on the 11s is being decreased - there's only one option which has that, which is D. (not to mention 11^22/11^11 is 11^11 so D is in the right 'range' for what you'd expect the exponent on 11 to be, as well)
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10 May 2018, 12:17
This is a very nice question. I spent lots of time trying to figure out the solution. The explanation give is very nice.
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Kindly press the +1Kudos if you like the explanation. Thanks a lot!!!

Re: M26-07   [#permalink] 10 May 2018, 12:17
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# M26-07

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