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# If x= 2^b - (8^8 + 8^6), for which of the following b values

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If x= 2^b - (8^8 + 8^6), for which of the following b values [#permalink]  20 Sep 2013, 10:12
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If x= 2^b - (8^8 + 8^6), for which of the following b values is x closest to 0?

(A) 20
(B) 24
(C) 25
(D) 30
(E) 42
[Reveal] Spoiler: OA

Last edited by Bunuel on 20 Sep 2013, 10:18, edited 1 time in total.
RENAMED THE TOPIC.
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Kudos [?]: 45048 [0], given: 6640

Re: If x= 2^b - (8^8 + 8^6), for which of the following b values [#permalink]  20 Sep 2013, 10:23
Expert's post
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abhisheksharma85 wrote:
If x= 2^b - (8^8 + 8^6), for which of the following b values is x closest to 0?

(A) 20
(B) 24
(C) 25
(D) 30
(E) 42

$$8^8 + 8^6=8^2*8^6 + 8^6=64*8^6+8^6=65*8^6=65*2^{18}\approx{2^{6}*2^{18}}=2^{24}$$.

$$2^b - (8^8 + 8^6)=0$$ --> $$2^b - 2^{24}=0$$ --> $$b=24$$

OR: 8^8 + 8^6 is very close to 8^8=2^24, thus b=24.
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Kudos [?]: 3077 [0], given: 38

Re: If x= 2^b - (8^8 + 8^6), for which of the following b values [#permalink]  20 Sep 2013, 10:27
Expert's post
abhisheksharma85 wrote:
If x= 2^b - (8^8 + 8^6), for which of the following b values is x closest to 0?

(A) 20
(B) 24
(C) 25
(D) 30
(E) 42

Dear abhisheksharma85,
I'm happy to help with this.

Recall the exponent rule:
(a^m)^n = a^(m*n)

Therefore,
8^8 + 8^6 = (2^3)^8 + (2^3)^6 = 2^24 + 2^18

This is the thing we want to cancel with 2^b. Well, keep in mind that 2^24 is 2^6 = 64 times larger than 2^18, so for the purposes of cancelling, we will ignore the tinier number. If we want to cancel the 2^24, we need b = 24, answer = (B).

Does all this make sense?
Mike
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Mike McGarry
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Posts: 28252
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Kudos [?]: 45048 [0], given: 6640

Re: If x= 2^b - (8^8 + 8^6), for which of the following b values [#permalink]  20 Sep 2013, 10:31
Expert's post
Bunuel wrote:
abhisheksharma85 wrote:
If x= 2^b - (8^8 + 8^6), for which of the following b values is x closest to 0?

(A) 20
(B) 24
(C) 25
(D) 30
(E) 42

$$8^8 + 8^6=8^2*8^6 + 8^6=64*8^6+8^6=65*8^6=65*2^{18}\approx{2^{6}*2^{18}}=2^{24}$$.

$$2^b - (8^8 + 8^6)=0$$ --> $$2^b - 2^{24}=0$$ --> $$b=24$$

OR: 8^8 + 8^6 is very close to 8^8=2^24, thus b=24.

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Hope it helps.
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Kudos [?]: 961 [0], given: 193

Re: If x= 2^b - (8^8 + 8^6), for which of the following b values [#permalink]  26 Feb 2014, 20:08
8^8 = 2^24
8^6 = 2^18

2^24 + 2^18 is nearly to 2^24, so b=24 = Answer = B
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Re: If x= 2^b - (8^8 + 8^6), for which of the following b values [#permalink]  16 Jun 2015, 02:12
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Re: If x= 2^b - (8^8 + 8^6), for which of the following b values   [#permalink] 16 Jun 2015, 02:12
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