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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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Math Expert
Joined: 02 Sep 2009
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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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10 Aug 2018, 01:23
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45% (medium)

Question Stats:

67% (01:42) correct 33% (02:10) wrong based on 621 sessions

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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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Math Expert
Joined: 02 Aug 2009
Posts: 7961
Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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10 Aug 2018, 02:03
4
1
3
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 $$4^x+4^{-x}+2=23+2=25$$
Sufficient

B
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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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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
B
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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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Updated on: 10 Aug 2018, 02:04
1
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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PKN

Rise above the storm, you will find the sunshine

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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10 Aug 2018, 02:04
2
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
VP
Joined: 09 Mar 2016
Posts: 1230
What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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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 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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12 Aug 2018, 04:09
1
$$4^x + 4^{−x} = 23$$

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

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

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 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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Posts: 1256
Location: India
GPA: 3.64
WE: Business Development (Energy and Utilities)
Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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12 Aug 2018, 04:21
1
dave13

See the approach
Attachment:

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)

_________________
VP
Joined: 09 Mar 2016
Posts: 1230
Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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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 [ 454.13 KiB | Viewed 6079 times ]

Retired Moderator
Joined: 27 Oct 2017
Posts: 1256
Location: India
GPA: 3.64
WE: Business Development (Energy and Utilities)
Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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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 [ 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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Senior Manager
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Posts: 420
Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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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 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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Posts: 1230
Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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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 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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12 Aug 2018, 08:04
1
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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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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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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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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11 Oct 2019, 05:20
Wouldn't root 25 yeild +-5 as two values making it insufficient ?
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Posts: 58390
Re: What is the value of 2^x + 2^(-x) ? (1) x < 0 (2) 4^x + 4^(−x) = 23  [#permalink]

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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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