If a and n are positive numbers, does 2a^(2x) = n ?

Question

Competition Mode Question

If a and n are positive numbers, does  2a^{2x}=n?

(1)  a^x+\frac{1}{a^x}=\sqrt{n+2}

(2)  x > 0

Are You Up For the Challenge: 700 Level Questions

CareerGeek · Answer

(1)  a^x+\frac{1}{a^x} = \sqrt{n+2} 
Squaring on both sides,
-->  (a^x+\frac{1}{a^x})^2 = n+2 
-->  a^{2x} + 2*a^x*\frac{1}{a^x} + \frac{1}{a^{2x}} = n + 2 
-->  a^{2x} + 2 + \frac{1}{a^{2x}} = n + 2 
-->  a^{2x} + \frac{1}{a^{2x}} = n 
Does  2a^{2x} = n  ?
-->  a^{2x} + \frac{1}{a^{2x}} = 2a^{2x} 
-->  a^{2x} = \frac{1}{a^{2x}} 
-->  a^{4x} = 1 
Only Possible if  x = 0 
But we DO NOT know the value of x -->  Insufficient

(2)  x > 0 
Does not talk anything about a & n -->  Insufficient

Combining (1) & (2),
If  x > 0 ,  a^{4x}  is equal to 1 when a = 1
And not equal to 1 when a is not equal to 1
 Insufficient

Option E

Archit3110 · Answer

#1
a^x+1/a^x=√n+2
possible when 
a=x=1 and n=2
we get yes sufficient
#2
x>0
a and n relation not know
insufficient
IMO A

exc4libur · Answer

a,n>0

(1)  a^x+1/a^x=\sqrt{n+2}    insufic

(a^x+1/a^x)^2=(\sqrt{n+2})^2…a^{2x}+a^{-2x}+2(a^x)(a^{-x})=n+2 
 a^{2x}+a^{-2x}+2(a^{x+(-x)})=n+2…a^{2x}+a^{-2x}+2(a^0)=n+2 
 a^{2x}+a^{-2x}+2=n+2…a^{2x}+a^{-2x}=n

(rephrase): 2a^{2x}=n…2a^{2x}=a^{2x}+a^{-2x} 
 2a^{2x}-a^{2x}=a^{-2x}…a^{2x}=a^{-2x} 
 a^{2x}=1/a^{2x}…a^{2x}a^{2x}=1…a^{4x}=1

(rephrase):a^{4x}=1?…yes:a=1…or…x=0

(2)  x>0    insufic

(1&2)   insufic

x>0…x≠0  but is  a=1 ?
 a=1,x>0:a^{4x}=1…1^{anything}=1…answer=yes 
 a=2,x>0:a^{4x}=1…2^{m4}≠1…answer=no

Ans (E)

freedom128 · Answer

Is  2a^{2x} = n ?

1) Squaring both sides, one will obtain:
 a^{2x} +a^{-2x}+2=n+2 
 a^{2x} +a^{-2x}=n

If x=0, then 1+1=2=n, then it must be true that  2a^{2x}= 2 = n .

If x>0, then  2a^{2x} =2n-2a^{-2x}  . Thus, it is not clear that  2a^{2x} = n . For clarity, consider a=1 or a=2.

NOT SUFFICIENT

2) We have no useful information on a and n. 
NOT SUFFICIENT

1)+2) Since x>0,  2a^{2x} =2n-2a^{-2x}  . Thus, it could be true (or false) that  2a^{2x} = n .
NOT SUFFICIENT

FINAL ANSWER IS (E)
   Kindly give kudos if you like the explanation

joohwangie · Answer

If a and n are positive numbers, does 2a^2x=n?

(1) a^x+1/(a^x)=root(n+2)
a^x+a^-x=root(n+2)
(a^x+a^-x)^2=n+2
a^2x+2a^x a^-x+a-2x=n+2
a^2x+1+a^-2x=n+2
a^(2x)+a^(-2x)-1=n
they not equal
sufficient

(2) x>0
we don't have any clue about a and n 
insufficient

Therefore, A

unraveled · Answer

If a and n are positive numbers, does  2a^{2x} = n ?

(1)  a^x+\frac{1}{a^x} = √(n+2) 
 a^{2x}+\frac{1}{a^{2x}} +2 = n+2 
 a^{2x}+\frac{1}{a^{2x}} = n 
a, x and n can take any value so

INSUFFICIENT.

(2) x > 0
None of the values are given.

INSUFFICIENT.

Together 1 and 2
INSUFFICIENT.

Answer E.

madgmat2019 · Answer

1) from this we won't be getting 2a^2x =n....... sufficient to say that they are not equal....but since they are positive no's...for a=1.....n becomes 2......for both of them....so insufficient

2) clearly insufficient

OA:E

debjit1990 · Answer

Ans: E

a)a^x + (1/a^x) = sqrt (n+2)
squaring both the sides
a^2x+(1/a^2x)+2=n+2
a^2x+(1/a^2x)=n
if x=0
n=2 and, 2a^(2x) = n
 x=0, so, 2a^0=2...true
if x=1,a=1,n=2 and equation holds..true
if x=1,a=2,n=17/4 and equation doesn't hold..false
not sufficient

b)x>0....alone clearly not sufficient

combined..for,x=1,a=1..true
 x=1,a=2..false

So, taking both not sufficient.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If a and n are positive numbers, does 2a^(2x) = n ?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards