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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

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
_________________

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)!

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

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15 May 2015, 10:29

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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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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.

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. \(2^{90}\).

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