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]

09 Jan 2013, 09:28

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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

[Reveal] Spoiler: OA

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: Absolute Value PS [#permalink]

09 Jan 2013, 10:33

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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]

09 Jan 2013, 11:07

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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]

09 Jan 2013, 12:12

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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] 09 Jan 2013, 12:12

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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

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

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