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# Given that x = 2^b – (8^30 + 16^5), which of the following values for

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Joined: 02 Sep 2009
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Given that x = 2^b – (8^30 + 16^5), which of the following values for  [#permalink]

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29 Feb 2016, 12:55
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Question Stats:

56% (01:32) correct 44% (01:40) wrong based on 395 sessions

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Given that x = 2^b – (8^30 + 16^5), which of the following values for b yields the lowest value for |x|?

A) 35
B) 90
C) 91
D) 95
E) 105

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Re: Given that x = 2^b – (8^30 + 16^5), which of the following values for  [#permalink]

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29 Feb 2016, 22:09
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Bunuel wrote:
Given that x = 2^b – (8^30 + 16^5), which of the following values for b yields the lowest value for |x|?

A) 35
B) 90
C) 91
D) 95
E) 105

$$x = 2^b - [8^{30} + 16^5]$$
$$= 2^b - [(2^3)^{30} + (2^4)^5]$$
$$=2^b - [(2^{90}) +(2^{20})]$$
$$= 2^b - [(2^{20})(2^{70} + 1)]$$
$$2^{70} + 1\approx2^{70}$$
therefore, $$x\approx2^b - [(2^{20})(2^{70})]$$
$$x\approx2^b - 2^{90}$$
for lowest value of $$|x|$$, $$b = 90$$
hence, B is the correct answer.

Alternatively,
$$x = 2^b - (8^{30} + 16^5)$$
$$= 2^b - [(2^3)^{30} + (2^4)^5]$$
$$= 2^b - [(2^{90}) +(2^{20})]$$
$$= 2^b - 2^{90} - 2^{20}$$
to find the lowest value of $$|x|$$, ignore the smaller exponent.
therefore, $$b = 90$$
hence, B is the correct answer.
##### General Discussion
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Joined: 17 Oct 2013
Posts: 54
Re: Given that x = 2^b – (8^30 + 16^5), which of the following values for  [#permalink]

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29 Feb 2016, 13:30
Bunuel wrote:
Given that x = 2^b – (8^30 + 16^5), which of the following values for b yields the lowest value for |x|?

A) 35
B) 90
C) 91
D) 95
E) 105

is answer E, since we have 2^b - 2^90 - 2^20, only 105 is closest to 110(90+20).
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Re: Given that x = 2^b – (8^30 + 16^5), which of the following values for  [#permalink]

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26 Jun 2017, 14:36
1
Bunuel wrote:
Given that x = 2^b – (8^30 + 16^5), which of the following values for b yields the lowest value for |x|?

A) 35
B) 90
C) 91
D) 95
E) 105

$$x = 2^b – (8^{30} + 16^5)$$

$$x = 2^b – (2^{3(30)} + 2^{4(5)})$$

$$x = 2^b – (2^{90} + 2^{20})$$

$$x = 2^b – (2^{20}(2^{70} + 1))$$

$$2^{20}(2^{70} + 1)$$ can be written as approximately $$= 2^{20}(2^{70}) = 2^{90}$$

$$x = 2^b - 2^{90}$$

Lowest value of $$|x|$$ is $$0$$. Hence $$b$$ should be equal to $$90$$.

$$x = 2^{90} - 2^{90}$$

$$2^b = 2^{90} . b = 90$$. Answer (B)...
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Re: Given that x = 2^b – (8^30 + 16^5), which of the following values for  [#permalink]

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02 Oct 2018, 03:14
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Re: Given that x = 2^b – (8^30 + 16^5), which of the following values for &nbs [#permalink] 02 Oct 2018, 03:14
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