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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  [#permalink]

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New post Updated on: 10 Jan 2013, 04:04
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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

Originally posted by sambam on 09 Jan 2013, 08:28.
Last edited by Bunuel on 10 Jan 2013, 04:04, 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  [#permalink]

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New post 23 Sep 2015, 04:34
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3
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  [#permalink]

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New post 09 Jan 2013, 10:07
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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  [#permalink]

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New post 09 Jan 2013, 09:33
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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  [#permalink]

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New post 09 Jan 2013, 11:12
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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  [#permalink]

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New post 29 Aug 2013, 02:28
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  [#permalink]

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New post 15 Sep 2013, 03:52
x=2^b-2^90-2^15 --> 2^15*(2^(b-15)-2^75) --> i wanna the result within the braces be 0, hence 2^(b-15)=2^75 --> b=90
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Re: Given that w = |x| and x = 2^b – (8^30 + 8^5), which of the  [#permalink]

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New post 16 Sep 2013, 09:42
5
1
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  [#permalink]

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New post 10 Jan 2014, 02:30
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 91

Now 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  [#permalink]

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New post 10 Jan 2014, 04:06
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  [#permalink]

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New post 23 Sep 2015, 00:04
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  [#permalink]

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