Given that w = |x| and x = 2^b – (8^30 + 8^5), which of the

Question

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

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

KarishmaB · Accepted Answer

This question has a few noteworthy points.
To get the smallest value of w (which is non negative), 2^b should be as close as possible to  (8^{30} + 8^5) .

8^{30} + 8^5) = (2^{90} + 2^{15})

Now a valid question is this: what is closer to  (2^{90} + 2^{15}) :  2^{90}  or  2^{91}  or higher powers? Let's focus on  2^{90}  and  2^{91}  only first.

Note a few things:  2^{91} = 2^{90} * 2^1

In other words, it is two times  2^{90}  i.e.  2^{90} + 2^{90}

So the question comes down to this: Is  (2^{90} + 2^{15})  closer to  2^{90} + 0  or  2^{90} + 2^{90}

Now, it is obvious that  2^{15}  will be much smaller than  2^{90} . 
 2^{15}  is equidistant from 0 and  2^{16}  on the number line (because using the same logic,  2^{16} = 2^{15} + 2^{15} ).
So  2^{15}  will be much closer to 0 compared with  2^{90} .

So  (2^{90} + 2^{15})  is closer to  2^{90} + 0  i.e.  2^{90} .

Hence, b must be 90.

Answer (B)

AbhiJ · Answer

The expression can be rewritten as  x=2^b - (2^{90} + 2^{15})

2^{90} >> 2^{15}  Hence expression becomes  x=2^b - (2^{90}) - a small quantity

Now we know that unless b = 90, the expression will have something of the order of  2^{90}  or even more.

Hence B.

Marcab · Answer

The  w=|x|  implies that we are not bothered about the sign.
The expression can be rewritten as  x=2^b - (2^{90} + 2^{15})

Now pick up the available answer choices.
For b=35,
 x=2^{35} - (2^{90} + 2^{15})  or  x=2^{35} - 2^{90}- 2^{15}  or  x= 2^{15} (2^{20} -2^{70} -1) . Since 1 is too less if compared to other available values, hence we neglect it. Now the expression becomes  x=2^{15}(2^{20}-2^{70})  or  x=2^{15} * 2^{20} * (-2^{50})

For b=90,
Same approach is applied and x comes out to be as  -2^{15} .

For b=91,
Same approach is applied and x comes out as  2^{15} * 2^{75}

For remaining answer choices, x would be even more.
Hence if b=90, we have the smallest value of  |w| .
hence +1B

PrashantPonde · Answer

Given,  x=2^b - (8^{30} + 8^{5}) 
i.e.  x= 2^b - (2^{90} + 2^{15}) = 2^b - 2^{15}(2^{75} + 1) = 2^b - 2^{15}(2^{75}) = 2^b - 2^{90}

PS: note that  (2^{75}+1)\approx{2^{75}}

Thus x is minimum when b is 90

Choice (B) is the correct answer!

ygdrasil24 · Answer

Can we afford to ignore 2^15 ?, even when options have more than 90 as answers

galiya · Answer

x=2^b-2^90-2^15 --> 2^15*(2^(b-15)-2^75) --> i wanna the result within the braces be 0, hence 2^(b-15)=2^75 --> b=90

shameekv · Answer

w = |x| means w will be lowest only when x = 0 since any non zero value for x, w will be positive and hence will be greater than 0 Hence, 2^b = (8^30 + 8^5) 2^b = 8^5(8^25 + 1) 2^b = 2^15 ( 2^75) [Neglecting 1 since 8^25 is much much greater than 1) Therefore 2^b =2^90 b = 90 ------- (b) Consider Kudos if it helped

Paris75 · Answer

Hi,

this is my process for this one:

2^b – (8^(30) + 8^(5)) ==> 2^b - (2^(90) + 2^(15))

Look at the answer choices. Eliminate all except those that are close to 2^90

You only have B and C:  b=90 or 91

Now look again at the question: is says  w = |x| . Coincidence? Never!

In fact you can have here a negative number because you have |x|.

Therefore bewteen 2^(15) (in fact it is - 2^(15) but as I said you are dealing with absolute value here so it is 2^(15)) and 2^(91) - 2^90 + 2^(15) which is the  smallest ?

2^(15) for sure (there is a huge difference here)!

Answer is therefore B.

Hope it helps!

PerfectScores · Answer

Backsolving will work wonders here:

If we start with any other number apart from 90

|2^something - 2^90 - 2^15| will always be greater than 2^15

Hence the only way it can be lowest i.e. 2^15 when b = 90 and 2^90 - 2^90 = 0

Hence answer is B

vityakim · Answer

This is how i thought about the problem.

Whenever i see big numbers such as 8^30, i assume that it should be simplified somehow, as we are not allowed to use calculator.
In thinking so, seeing 2^b is a relief because 8^30 can be written as 2^3^30 = 2^90

So we get, x= 2^b - (2^90 + 2^15)
Afterwards, plug choices.

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

Saks116 · Answer

Why we aren’t considering the exponents property of multiplication : a^m + a^n = a^m+n

Posted from my mobile device

Bunuel · Answer

Not correct. It's  a^m * a^n = a^{(m+n)} .

8. Exponents and Roots of Numbers 
     Theory   
 Exponents and Roots on the GMAT: Tips and hints 
 Cyclicity and the remainders on the GMAT 
     Questions   
 Units digits, exponents, remainders problems 
 Tough and tricky DS exponents and roots questions with detailed solutions 
 Tough and tricky PS exponents and roots questions with detailed solutions 
 Exponents PS Questions 
 Roots DS Questions 
 Roots PS Questions

Check below for more:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

Hope this helps.

bumpbot · Answer

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Given that w = |x| and x = 2^b – (8^30 + 8^5), which of the

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