If x = 2^b – (2^20+ 2^29), for which of the following b values is x

Question

If x = 2^b – (2^20+ 2^29), for which of the following b values is x closest to zero?

(A) 20
(B) 24
(C) 25
(D) 29
(E) 42

Bunuel · Accepted Answer

2^{20}+ 2^{29} = 2^{20}(1+ 2^{9})\approx 2^{20}2^{9}=2^{29}

2^b – (2^{20}+ 2^{29})=0 ;

2^b – 2^{29}=0 ;

2^b = 2^{29} ;

b = 29 .

Answer: D.

Bunuel · Answer

Kinshook · Answer

Asked: If x = 2^b – (2^20+ 2^29), for which of the following b values is x closest to zero?

2^20 = 1024*1024 = 1048576
2^29 = 1024 * 1024* 512 = 512*1048576
2^20 + 2^29 = 513*1048576 ~ 512*1048576 ~ 2^29

IMO D

sanjna2023 · Answer

Hi  Bunuel ,

Unsure how you approximated the value in Step 1? What happens to 2^20?

Thanks

Bunuel · Answer

2^{20}2^{9}=2^{20+9}=2^{29}

GMATWhizTeam · Answer

This question tests your ability to factor out algebraic expressions and use the laws of exponents to simplify. It also tests your estimation skills.

First, let us try to deal with the expression inside the parenthesis, i.e. ( 2^{20}  +  2^{29} )

The term  2^{20}  can be factored out and hence,

( 2^{20}  +  2^{29} ) =  2^{20}  (1 +  2^9 )

This is the stage where we need to estimate. Clearly, 1 is very small compared to  2^9  and hence, there’s nothing wrong in saying that (1 +  2^9 ) can be approximated to  2^9

Therefore, approximate value of ( 2^{20}  +  2^{29} ) =  2^{20}  *  2^9  =  2^{29}  (using the laws of exponents)

If x has to be zero,  2^b  and  2^{29}  should cancel out; this means, the value of b has to be 29.

The correct answer option is D.

bumpbot · Answer

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If x = 2^b – (2^20+ 2^29), for which of the following b values is x

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If x = 2^b – (2^20+ 2^29), for which of the following b values is x

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