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

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If a and n are positive numbers, does 2a^{2x}=n? [#permalink]  03 Mar 2013, 14:44
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If a and n are positive numbers, does $$2a^{2x}=n?$$

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

(2) $$x > 0$$
[Reveal] Spoiler: OA

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Re: If a and n are positive numbers, does 2a^{2x}=n? [#permalink]  03 Mar 2013, 15:22
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PraPon wrote:
If a and n are positive numbers, does $$2a^{2x}=n?$$

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

(2) $$x > 0$$

(1) $$a^x+\frac{1}{a^x}=\sqrt{n+2}$$ ALONE when squarred : $$( a^{x} + a^{-x} )^2 = n+2$$

so, $$n = (a^{x} + a^{-x})^{2} - 2$$

$$n = a^{2x}+a^{-2x}+2a^{x}*a^{-x}-2$$ ($$a^{x}*a^{-x}=1$$)

$$n = a^{2x}+a^{-2x}$$

back to the question :

$$2a^{2x}-n = 2a^{2x}-a^{2x}-a^{-2x}$$
$$2a^{2x}-n = a^{2x}-a^{-2x}$$

Hence, A insufficent

(2) $$x > 0$$ ALONE

Clearly insufficent

both (1) and (2) INSUFF

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Re: If a and n are positive numbers, does 2a^{2x}=n? [#permalink]  03 Mar 2013, 21:53
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Expert's post
PraPon wrote:
If a and n are positive numbers, does $$2a^{2x}=n?$$

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

(2) $$x > 0$$

We know that $$(a^x+a^-x)^2 = a^2x+a^-2x+2.$$

From F.S 1, we have (n+2) =$$a^2x+a^-2x+2$$

or n =$$a^2x+a^-2x$$. Thus, the question stem is asking whether $$2a^2x$$= n?

If it has to be true, then $$a^2x+a^-2x = 2a^2x$$.

or a^(4x) = 1. Now, for x=0, we get a YES. But depending on the values of a and x, this will change. Thus, insufficient.

From F.S 2, we only have x>0. Clearly Insufficient.

Combining both, we know that x is not equal to zero. However, if a=1, we can still get a^4x = 1. Insufficient.

E.
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Re: If a and n are positive numbers, does 2a^{2x}=n? [#permalink]  12 May 2014, 21:12
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Re: If a and n are positive numbers, does 2a^{2x}=n?   [#permalink] 12 May 2014, 21:12
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