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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]

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New post Updated on: 20 Sep 2013, 11:18
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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

Originally posted by abhisheksharma85 on 20 Sep 2013, 11:12.
Last edited by Bunuel on 20 Sep 2013, 11:18, edited 1 time in total.
RENAMED THE TOPIC.
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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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New post 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


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

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

Answer: B.

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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New post 20 Sep 2013, 11:27
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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

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

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New post 20 Sep 2013, 11:31
1
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


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

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

Answer: B.

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


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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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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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New post 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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New post 01 Sep 2017, 18:28
1
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 \(= 24\)

Answer (B)...
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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