Equations with three variables

[Milovan]

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

(1) a^x + 1/a^x = sqrt (n+2)

(2) x > 0

This question is from Kaplan Prep test. Explanation is very strange and I am not able to crack it completely. Can someone help?

[batwaraanirudh]

The question stem states: a and n are positive integers and it is asking whether (2a)^2x = n

Lets look at statement 2 first:

The information given is only about x and nothing is mentioned about a or n. With only this information we would not be able to comment anything about the relationship between a, x and n. Therefore 2 is INSUFFICIENT

Now lets have a look at statement 1:

a^x + 1/a^x = sqrt(n+2)
Lets square both the sides
=> (a^x + 1/a^x)^2 = (sqrt(n+2))^2
=>a^2x + 1/a^2x + 2 = n+2
=>a^2x + 1/a^2x = n

This is the best we can do with this statement. again we are not in a certain position to comment if 2a^2x = n+2.

If a=1, then the statement is correct else it is false.
Since we have no information about a, Therefore 1 alone is NOT SUFFICIENT

Lets look at both the statements combined:
We get information about x from this statement but not about a. so both statements together is also NOT SUFFICIENT.

Hence the answer is E

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Hi Milovan,

Does the original question state that the variables a and n are "positive integers" or "positive numbers." The wording here is important.

GMAT assassins aren't born, they're made,
Rich

[GMATGuruNY]

I believe that the question stem should read as follows:

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

(1) a^x + 1/a^x = sqrt (n+2)

(2) x > 0

Remember the following identity:
(x+y)² = x² + 2xy + y²

Statement 1: a^x + 1/a^x = sqrt (n+2)
a^x + a^-x = √(n+2).

Squaring both sides, we get:
(a^x)² + 2(a^x)(a^-x) + (a^-x)² = n+2
a^2x + 2(a�) + a^(-2x) = n+2
a^2x + a^(-2x) = n.

Substituting a^2x + a^(-2x) = n into the question stem -- Does 2(a^2x) = n? -- we get:
Does 2(a^2x) = a^2x + a^(-2x)?
Does a^2x = a^(-2x)?
Does a^4x = 1?

Case 1: If a=1 and x=1, the answer is YES.
Case 2: If a=2 and x=1, the answer is NO.
INSUFFICIENT.

Cases 1 and 2 also satisfy statement 2.
Thus, even when the statements are combined, the answer to the question stem can be YES or NO.

The correct answer is E.

[Milovan]

Thanks a lot Mitch, now it is totally clear. I have no idea why this example was so hard for me to grasp even though I knew it is not so complicated.