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Given that w = x and x = 2^b – (8^30 + 8^5), which of the
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Given that w = x and x = 2^b – (8^30 + 8^5), which of the following values for b yields the lowest value for w? (A) 35 (B) 90 (C) 91 (D) 95 (E) 105
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Originally posted by sambam on 09 Jan 2013, 09:28.
Last edited by Bunuel on 10 Jan 2013, 05:04, edited 1 time in total.
Renamed the topic and edited the question.




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Re: Given that w = x and x = 2^b – (8^30 + 8^5), which of the
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23 Sep 2015, 05:34
sambam wrote: Given that w = x and x = 2^b – (8^30 + 8^5), which of the following values for b yields the lowest value for w?
(A) 35 (B) 90 (C) 91 (D) 95 (E) 105 This question has a few noteworthy points. To get the smallest value of w (which is non negative), 2^b should be as close as possible to \((8^{30} + 8^5)\). \(8^{30} + 8^5) = (2^{90} + 2^{15})\) Now a valid question is this: what is closer to \((2^{90} + 2^{15})\): \(2^{90}\) or \(2^{91}\) or higher powers? Let's focus on \(2^{90}\) and \(2^{91}\) only first. Note a few things: \(2^{91} = 2^{90} * 2^1\) In other words, it is two times \(2^{90}\) i.e. \(2^{90} + 2^{90}\) So the question comes down to this: Is \((2^{90} + 2^{15})\) closer to \(2^{90} + 0\) or \(2^{90} + 2^{90}\) Now, it is obvious that \(2^{15}\) will be much smaller than \(2^{90}\). \(2^{15}\) is equidistant from 0 and \(2^{16}\) on the number line (because using the same logic, \(2^{16} = 2^{15} + 2^{15}\)). So \(2^{15}\) will be much closer to 0 compared with \(2^{90}\). So \((2^{90} + 2^{15})\) is closer to \(2^{90} + 0\) i.e. \(2^{90}\). Hence, b must be 90. Answer (B)
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Re: Absolute Value PS
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09 Jan 2013, 11:07
The expression can be rewritten as \(x=2^b  (2^{90} + 2^{15})\) \(2^{90} >> 2^{15}\) Hence expression becomes \(x=2^b  (2^{90})  a small quantity\) Now we know that unless b = 90, the expression will have something of the order of \(2^{90}\) or even more. Hence B.
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Re: Absolute Value PS
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09 Jan 2013, 10:33
sambam wrote: Given that w = x and x = 2^b – (8^30 + 8^5), which of the following values for b yields the lowest value for w?
(A) 35 (B) 90 (C) 91 (D) 95 (E) 105
The \(w=x\) implies that we are not bothered about the sign. The expression can be rewritten as \(x=2^b  (2^{90} + 2^{15})\) Now pick up the available answer choices. For b=35, \(x=2^{35}  (2^{90} + 2^{15})\) or \(x=2^{35}  2^{90} 2^{15}\) or \(x= 2^{15} (2^{20} 2^{70} 1)\). Since 1 is too less if compared to other available values, hence we neglect it. Now the expression becomes \(x=2^{15}(2^{20}2^{70})\) or \(x=2^{15} * 2^{20} * (2^{50})\) For b=90, Same approach is applied and x comes out to be as \(2^{15}\). For b=91, Same approach is applied and x comes out as \(2^{15} * 2^{75}\) For remaining answer choices, x would be even more. Hence if b=90, we have the smallest value of \(w\). hence +1B
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Re: Absolute Value PS
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09 Jan 2013, 12:12
Given, \(x=2^b  (8^{30} + 8^{5})\) i.e. \(x= 2^b  (2^{90} + 2^{15}) = 2^b  2^{15}(2^{75} + 1) = 2^b  2^{15}(2^{75}) = 2^b  2^{90}\) PS: note that \((2^{75}+1)\approx{2^{75}}\) Thus x is minimum when b is 90 Choice (B) is the correct answer!
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Re: Absolute Value PS
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29 Aug 2013, 03:28
PraPon wrote: Given, \(x=2^b  (8^{30} + 8^{5})\) i.e. \(x= 2^b  (2^{90} + 2^{15}) = 2^b  2^{15}(2^{75} + 1) = 2^b  2^{15}(2^{75}) = 2^b  2^{90}\)
PS: note that \((2^{75}+1)\approx{2^{75}}\)
Thus x is minimum when b is 90
Choice (B) is the correct answer! Can we afford to ignore 2^15 ?, even when options have more than 90 as answers



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Re: Given that w = x and x = 2^b – (8^30 + 8^5), which of the
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15 Sep 2013, 04:52
x=2^b2^902^15 > 2^15*(2^(b15)2^75) > i wanna the result within the braces be 0, hence 2^(b15)=2^75 > b=90



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Re: Given that w = x and x = 2^b – (8^30 + 8^5), which of the
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16 Sep 2013, 10:42
sambam wrote: Given that w = x and x = 2^b – (8^30 + 8^5), which of the following values for b yields the lowest value for w?
(A) 35 (B) 90 (C) 91 (D) 95 (E) 105 w = x means w will be lowest only when x = 0 since any non zero value for x, w will be positive and hence will be greater than 0 Hence, 2^b = (8^30 + 8^5) 2^b = 8^5(8^25 + 1) 2^b = 2^15 ( 2^75) [Neglecting 1 since 8^25 is much much greater than 1) Therefore 2^b =2^90 b = 90  (b) Consider Kudos if it helped



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Re: Given that w = x and x = 2^b – (8^30 + 8^5), which of the
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10 Jan 2014, 03:30
Hi, this is my process for this one: 2^b – (8^(30) + 8^(5)) ==> 2^b  (2^(90) + 2^(15)) Look at the answer choices. Eliminate all except those that are close to 2^90 You only have B and C: b=90 or 91Now look again at the question: is says \(w = x\). Coincidence? Never! In fact you can have here a negative number because you have x. Therefore bewteen 2^(15) (in fact it is  2^(15) but as I said you are dealing with absolute value here so it is 2^(15)) and 2^(91)  2^90 + 2^(15) which is the smallest? 2^(15) for sure (there is a huge difference here)! Answer is therefore B. Hope it helps!
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Re: Given that w = x and x = 2^b – (8^30 + 8^5), which of the
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10 Jan 2014, 05:06
sambam wrote: Given that w = x and x = 2^b – (8^30 + 8^5), which of the following values for b yields the lowest value for w?
(A) 35 (B) 90 (C) 91 (D) 95 (E) 105 Backsolving will work wonders here: If we start with any other number apart from 90 2^something  2^90  2^15 will always be greater than 2^15 Hence the only way it can be lowest i.e. 2^15 when b = 90 and 2^90  2^90 = 0 Hence answer is B
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Re: Given that w = x and x = 2^b – (8^30 + 8^5), which of the
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23 Sep 2015, 01:04
This is how i thought about the problem.
Whenever i see big numbers such as 8^30, i assume that it should be simplified somehow, as we are not allowed to use calculator. In thinking so, seeing 2^b is a relief because 8^30 can be written as 2^3^30 = 2^90
So we get, x= 2^b  (2^90 + 2^15) Afterwards, plug choices.
(A) 35 (B) 90 (C) 91 (D) 95 (E) 105



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Re: Given that w = x and x = 2^b – (8^30 + 8^5), which of the
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Re: Given that w = x and x = 2^b – (8^30 + 8^5), which of the
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