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

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Joined: 17 Jun 2013
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If x= 2^b - (8^8 + 8^6), for which of the following b values [#permalink]

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20 Sep 2013, 11:12
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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, 11:18, edited 1 time in total.
RENAMED THE TOPIC.
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Joined: 02 Sep 2009
Posts: 44399
Re: If x= 2^b - (8^8 + 8^6), for which of the following b values [#permalink]

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20 Sep 2013, 11:23
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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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Re: If x= 2^b - (8^8 + 8^6), for which of the following b values [#permalink]

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20 Sep 2013, 11:27
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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Math Expert
Joined: 02 Sep 2009
Posts: 44399
Re: If x= 2^b - (8^8 + 8^6), for which of the following b values [#permalink]

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20 Sep 2013, 11:31
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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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Re: If x= 2^b - (8^8 + 8^6), for which of the following b values [#permalink]

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26 Feb 2014, 21:08
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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If x= 2^b - (8^8 + 8^6), for which of the following b values [#permalink]

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01 Sep 2017, 18:28
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

$$x= 2^b - (8^8 + 8^6)$$

$$8^8 + 8^6 = 8^6(8^2+1)$$

We can approximate the value. $$1$$ is very small value compared to $$8^2$$ hence we can ignore $$1$$. Therefore we get;

$$8^8 + 8^6 = 8^6(8^2) = 8^{(6+2)} = 8^8 = (2^3)^{8} = 2^ {(3*8)} = 2^{24}$$

$$x= 2^b - 2^{24}$$

For "$$x$$" to be closest to $$0$$, value of "$$b$$" should be $$= 24$$

If x= 2^b - (8^8 + 8^6), for which of the following b values   [#permalink] 01 Sep 2017, 18:28
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