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What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23

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What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 10 Aug 2018, 01:23
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Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 10 Aug 2018, 02:03
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What is the value of 2^x + 2^(-x) ?

\(2^x+2^{-x}=2^x+\frac{1}{2^x}\)
Square both sides..
\(2^{2x}+2^{-2x}+2=4^x+4^{-x}+2\)

(1) x < 0
Does not effect the equation at all..
Insufficient

(2) 4^x + 4^(−x) = 23
So \(4^x+4^{-x}+2=23+2=25\)
Our answer is √25=5
Sufficient

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What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 10 Aug 2018, 01:47
(1) Insufficient
(2)
4^x + 4^(-x) = 23
<=> [4^x + 4^(-x) + 2*2^x*2^(-x)] - 2*2^x*2^(-x) = 23
<=> (2^x + 2^(-x))^2 - 2*2^x*2^(-x) = 23
<=> (2^x + 2^(-x))^2 = 25 (Because 2^x*2^(-x) = 1)
<=> 2^x + 2^(-x) = 5
=> Sufficient
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What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post Updated on: 10 Aug 2018, 02:04
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2
Bunuel wrote:
What is the value of 2^x + 2^(-x) ?

(1) x < 0
(2) 4^x + 4^(−x) = 23


NEW question from GMAT® Quantitative Review 2019


(DS05989)


St1:- x < 0

a) when x = -1, then \(2^x+2^{-x}=2^{-1}+2^{1}=\frac{1}{2}+2=\frac{5}{2}\)
b) when x=-2, then \(2^x+2^{-x}=2^{-2}+2^{2}=\frac{1}{4}+4=\frac{17}{4}\)

Insufficient.

St2:- \(4^x + 4^{−x} = 23\)
Now we have to find out a relation between question stem and statement(2), otherwise it is hard to determine question stem.
\((2^x+2^{-x})^2=2^{2x}+2*2^x*2^{-x}+2^{-2x}=(2^2)^x+2^{x+(-x)+1}+(2^2)^{-x}=4^x+2^1+4^{-x}=4^x+4^{-x}+2\)

So, \((2^x+2^{-x})^2=23+2=25\)
Or, \(2^x+2^{-x}=\sqrt{25}=5\)

N.B:- \(2^x+2^{-x}\) is always positive for all value of x. So, -5 is not acceptable.

Sufficient.

Ans. (B)
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Originally posted by PKN on 10 Aug 2018, 02:02.
Last edited by PKN on 10 Aug 2018, 02:04, edited 1 time in total.
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What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 10 Aug 2018, 02:04
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1
1
Bunuel wrote:
What is the value of 2^x + 2^(-x) ?

(1) x < 0
(2) 4^x + 4^(−x) = 23


NEW question from GMAT® Quantitative Review 2019


(DS05989)


OA: B
What is the value of \(2^x + 2^{-x}\) ?
or What is value of \(x\)?

(1) : \(x < 0\)
plugging \(x =-1\)
\(2^x + 2^{-x} = 2^{-1}+2^{1}= 0.5 + 2 =2.5\)
plugging \(x =-2\)
\(2^x + 2^{-x} = 2^{-2}+2^{2}= \frac{1}{4} + 4 = 0.25 + 4 = 4.25\)
We are not getting a unique value of \(2^x + 2^{-x}\)
So Statement 1 alone is not sufficient

(2) \(4^x + 4^{−x} = 23\)
Let \(2^x + 2^{-x}=y\)
Squaring both sides, we get
\((2^x + 2^{-x})^2=y^2\)
\((2^x)^2 +(2^{-x})^2 + 2(2^x)(2^{-x}) =y^2\)
\(2^x.2^x +2^{-x}.2^{-x} + 2 =y^2\)
\(4^x + 4^{−x} + 2 =y^2\)
\(23 + 2 =y^2\)
\(y^2=25 ; y=+5,-5\) (\(-5\) rejected as minimum value of \(2^x + 2^{-x}\) is \(2\) at \(x =0\))
using A.M≥ G.M
\(\frac{2^x + 2^{-x}}{2}≥\sqrt{{2^x * 2^{-x}}}\)
\(2^x + 2^{-x}≥2\)
So Statement 2 alone is sufficient
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What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 12 Aug 2018, 03:29
1
Bunuel wrote:
What is the value of 2^x + 2^(-x) ?

(1) x < 0
(2) 4^x + 4^(−x) = 23


NEW question from GMAT® Quantitative Review 2019


(DS05989)



statement two. detailed WRONG approach :grin: where did i go wrong guys ? can anyone comment please ? :-)

\(4^x + 4^{−x} = 23\)

\(2^{2x} + 2^{2 *(−x)} = 23\) equate bases

\(2x+(-2x) = 23\) now what ? :?

gmatbusters can you explain this for me please :-)
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What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 12 Aug 2018, 04:09
1
Your Query :
\(4^x + 4^{−x} = 23\)

\(2^{2x} + 2^{2 *(−x)} = 23\) equate bases

\(2x+(-2x) = 23\) now what ? :?- YOUR APPROACH IS INVALID, Even I am confused, what u did here :dazed

Response:

if a^b = a^c
then by equating base , we can say b = c
, ONLY IF
a is not equal to 0, 1, -1

for example: 1^4= 1^98, but 4 is not equal to 98.

dave13 wrote:
Bunuel wrote:
What is the value of 2^x + 2^(-x) ?

(1) x < 0
(2) 4^x + 4^(−x) = 23


NEW question from GMAT® Quantitative Review 2019


(DS05989)



statement two. detailed WRONG approach :grin: where did i go wrong guys ? can anyone comment please ? :-)

\(4^x + 4^{−x} = 23\)

\(2^{2x} + 2^{2 *(−x)} = 23\) equate bases

\(2x+(-2x) = 23\) now what ? :?

gmatbusters can you explain this for me please :-)

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Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 12 Aug 2018, 04:21
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dave13

See the approach
Attachment:
WhatsApp Image 2018-08-12 at 16.48.05.jpeg
WhatsApp Image 2018-08-12 at 16.48.05.jpeg [ 75.32 KiB | Viewed 6128 times ]


Bunuel wrote:
What is the value of 2^x + 2^(-x) ?

(1) x < 0
(2) 4^x + 4^(−x) = 23


NEW question from GMAT® Quantitative Review 2019


(DS05989)

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Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 12 Aug 2018, 05:09
gmatbusters wrote:
dave13

See the approach
Attachment:
The attachment WhatsApp Image 2018-08-12 at 16.48.05.jpeg is no longer available


Bunuel wrote:
What is the value of 2^x + 2^(-x) ?

(1) x < 0
(2) 4^x + 4^(−x) = 23


NEW question from GMAT® Quantitative Review 2019


(DS05989)


gmatbusters many thanks for explanation i have some tech questions :) can you please explain how you got values marked in green :-)
Attachments

WhatsApp%20Image%202018-08-12%20at%2016.48.05.jpeg_LI.jpg
WhatsApp%20Image%202018-08-12%20at%2016.48.05.jpeg_LI.jpg [ 454.13 KiB | Viewed 6079 times ]

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Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 12 Aug 2018, 05:34
1
dave13, I used it to complete the square formula.

See this, Here I have found the Square of the Expression in the Question stem,
you will find it easier.
Attachment:
WhatsApp Image 2018-08-12 at 18.00.58.jpeg
WhatsApp Image 2018-08-12 at 18.00.58.jpeg [ 87.79 KiB | Viewed 6051 times ]


dave13 wrote:
gmatbusters wrote:
dave13

See the approach
Attachment:
The attachment WhatsApp Image 2018-08-12 at 16.48.05.jpeg is no longer available


Bunuel wrote:
What is the value of 2^x + 2^(-x) ?

(1) x < 0
(2) 4^x + 4^(−x) = 23


NEW question from GMAT® Quantitative Review 2019


(DS05989)


gmatbusters many thanks for explanation i have some tech questions :) can you please explain how you got values marked in green :-)

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Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 12 Aug 2018, 06:25
1
dave13 wrote:
Bunuel wrote:
What is the value of 2^x + 2^(-x) ?

(1) x < 0
(2) 4^x + 4^(−x) = 23


NEW question from GMAT® Quantitative Review 2019


(DS05989)



statement two. detailed WRONG approach :grin: where did i go wrong guys ? can anyone comment please ? :-)

\(4^x + 4^{−x} = 23\)

\(2^{2x} + 2^{-2x} = 23\) equate bases

\(2x+(-2x) = 23\) now what ? :?

gmatbusters can you explain this for me please :-)


dave13
\(2^{2x} + 2^{-2x} = 23\)

\(2^x*2^x +2^{-x}*2^{-x}= 23\)

\((2^x)^2+(2^{-x})^2=23\) .....(1)

Now Using \(a^2 +b^2 +2ab =(a+b)^2\)
\(a^2 +b^2 =(a+b)^2 - 2ab\)

here \(a = 2^x ; b =2^{-x}\)

\((2^x)^2+(2^{-x})^2 =(2^x + 2^{-x})^2 - 2*2^x*2^{-x} =(2^x + 2^{-x})^2 - 2*2^{x-x}=(2^x + 2^{-x})^2 - 2*2^0=(2^x + 2^{-x})^2 - 2\)

\((2^x)^2+(2^{-x})^2=(2^x + 2^{-x})^2 - 2\).........(2)

Substituting value of \((2^x)^2+(2^{-x})^2\) from (2) into (1) , we get

\((2^x + 2^{-x})^2 - 2=23\)

\((2^x + 2^{-x})^2=25\)

I hope it is clear now
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Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 12 Aug 2018, 07:54
Princ wrote:
dave13 wrote:
Bunuel wrote:
What is the value of 2^x + 2^(-x) ?

(1) x < 0
(2) 4^x + 4^(−x) = 23


NEW question from GMAT® Quantitative Review 2019


(DS05989)



statement two. detailed WRONG approach :grin: where did i go wrong guys ? can anyone comment please ? :-)

\(4^x + 4^{−x} = 23\)

\(2^{2x} + 2^{-2x} = 23\) equate bases

\(2x+(-2x) = 23\) now what ? :?

gmatbusters can you explain this for me please :-)


dave13
\(2^{2x} + 2^{-2x} = 23\)

\(2^x*2^x +2^{-x}*2^{-x}= 23\)

\((2^x)^2+(2^{-x})^2=23\) .....(1)

Now Using \(a^2 +b^2 +2ab =(a+b)^2\)
\(a^2 +b^2 =(a+b)^2 - 2ab\)

here \(a = 2^x ; b =2^{-x}\)

\((2^x)^2+(2^{-x})^2 =(2^x + 2^{-x})^2 - 2*2^x*2^{-x} =(2^x + 2^{-x})^2 - 2*2^{x-x}=(2^x + 2^{-x})^2 - 2*2^0=(2^x + 2^{-x})^2 - 2\)

\((2^x)^2+(2^{-x})^2=(2^x + 2^{-x})^2 - 2\).........(2)

Substituting value of \((2^x)^2+(2^{-x})^2\) from (2) into (1) , we get

\((2^x + 2^{-x})^2 - 2=23\)

\((2^x + 2^{-x})^2=25\)

I hope it is clear now


Princ many thanks for taking to to explain. i have one question: for instance this formula \(a^2 +b^2 +2ab =(a+b)^2\) i know, but this formula \(a^2 +b^2 =(a+b)^2 - 2ab\) i didnt know :)

can you please explain what is the relationship between these two formulas :-) \(a^2 +b^2 +2ab =(a+b)^2\) , \(a^2 +b^2 =(a+b)^2 - 2ab\) are these two different formulas ? :?

thank you:)
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Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 12 Aug 2018, 08:04
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Quote:
Princ many thanks for taking to to explain. i have one question: for instance this formula \(a^2 +b^2 +2ab =(a+b)^2\) i know, but this formula \(a^2 +b^2 =(a+b)^2 - 2ab\) i didnt know :)

can you please explain what is the relationship between these two formulas :-) \(a^2 +b^2 +2ab =(a+b)^2\) , \(a^2 +b^2 =(a+b)^2 - 2ab\) are these two different formulas ? :?

thank you:)


dave13

These are the same formula
as you know that
\(a^2+b^2+2ab = (a+b)^2\)

Substracting \(2ab\) from both sides,we get

\(a^2+b^2+2ab-2ab = (a+b)^2-2ab\)
\(a^2+b^2= (a+b)^2-2ab\)
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Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 16 Oct 2018, 01:10
gmatbusters why 5 why not +- 5?
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Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 16 Oct 2018, 01:20
Hi
If a is positive, than \(a^x\) can never be negative.
(Min value would tend to 0, when X is negative infinity.)
Hence\(2^x\) is always positive.
I hope it resolves your query. Feel free to tag me again.

manjot123 wrote:
gmatbusters why 5 why not +- 5?

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Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 20 May 2019, 00:59
is knowing a2+b2=(a+b)2−2ab the only way of solving this question? is there any other alternative to this, incase this doesnt pop up the first thing i see question like this on exam?
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Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 11 Oct 2019, 05:20
Wouldn't root 25 yeild +-5 as two values making it insufficient ?
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Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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New post 11 Oct 2019, 05:34
Nityanshu1990 wrote:
Wouldn't root 25 yeild +-5 as two values making it insufficient ?


When the GMAT provides the square root sign for an even root, such as \(\sqrt{x}\) or \(\sqrt[4]{x}\), then the only accepted answer is the positive root. Even roots have only a positive value on the GMAT. That is, \(\sqrt{25}=5\), NOT +5 or -5.

In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5.
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Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23   [#permalink] 11 Oct 2019, 05:34
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