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Quote:
If a and n are positive numbers, does \(2a^{2x}=n\)?

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

\(a,n>0\)

(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)
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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)
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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
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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.
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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
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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.
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