Question: What is the value of 2^x + 2^(-x)?

(1) x < 0
(2) 4^x + 4^(−x) = 23

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

“What is the value of 2^x + 2^(-x)” is a topic of the GMAT Quantitative reasoning section of GMAT. The questions of GMAT Data Sufficiency come up with a problem statement and two factual statements. This specific GMAT data sufficiency question evaluates the candidate’s level of efficiency in interpreting graphical data and solving quantitative problems. The hardest portion of these types of questions mainly comes from clever or obtuse wording that candidates usually ignore. GMAT Quant section consists of a total of 31 questions. Out of which, GMAT data sufficiency includes 15 questions which are two-fifths of the entire 31 GMAT quant questions

Solution and Explanation:

Approach Solution 1:

The problem statement asks to find the value of 2^x + 2^(-x)
Therefore, we can write it as, 2^x + 2^(-x) = 2^x + 1/2^x

By squaring both sides, we get,
2^2x + 2^(-2x) + 2 = 4^x + 4^(-x) + 2

1. The statement suggests that the value of x is less than zero. The argument of this statement does not support the equation derived from the question.
   Therefore, this statement alone is not sufficient to find the value of the equation cited in the question.
2. The statement suggests that 4^x + 4^(−x) = 23.
   Therefore by putting the value in the equation derived from the question, we can get,
   4^x + 4^(-x) + 2 = 23 + 2 =25
   Therefore, 2^x + 2^(-x) = √25 = 5

Hence statement (2) alone is sufficient to find the value of the derived equation of the question.

Correct Answer: (B)

Approach Solution 2:

Here in this question, the problem statement asks to find the value of 2^x + 2^(-x)

1. Statement (1) suggests the value of x is less than zero.

Therefore, when the value of x is -1, then we get,
2^x+2^(−x) = 2^(−1) + 2^1 = ½ + 2 = 5/2

Therefore, when the value of x is -2, then we get,
2^x+2^(−x) = 2^(−2) + 2^2= ¼ + 4 = 17/4

We are not obtaining a unique value of x.
Thus, statement (1) alone is not sufficient to find the value of 2^x + 2^(-x).

2. Statement (2) cites 4^x + 4^(−x) = 23

Now, it is required to establish a relation between the problem statement and the statement (2).
The given equation in the question states 2^x + 2^(-x)
By squaring the equation, we get,

(2^x + 2^-x)^2 = 2^2x + 2∗2^x∗2^−x + 2^−2x
                         =2^2x +2^{x+(−x)+1}+2^− 2x   
                         =4^x + 2^1 + 4^−x
                         =4^x + 4^−x + 2

Therefore, (2^x + 2^-x)^2 = 23 + 2 = 25
Or, 2^x + 2^-x = √25 = 5

The value of 2^x + 2^-x is always positive for all values of x. Henceforth, the negative value of x = -5 is not acceptable.
Thus statement (2) alone is sufficient to find the value of the derived equation of the question.

Correct Answer: (B)

Approach Solution 3:

The problem statement asks to find the value of 2^x + 2^(-x)

1. This statement suggests that the value of x is less than zero.
   Therefore, putting x = -1, we get,
   2^x+2^(−x) = 2^(−1) + 2^1 = 0.5 + 2 = 2.5
   Therefore, putting x = -2, we get,
   2^x+2^(−x) = 2^(−2) + 2^2= ¼ + 4 = 0.25 + 4 = 4.25
   We are not receiving a unique value of x.
   Thus, statement (1) alone is not sufficient to find the value of 2^x + 2^(-x).
2. This statement states 4^x + 4^(−x) = 23
   Let, 2^x + 2^(-x) = y
   By squaring both sides of the equation, we get,
   (2^x + 2^-x)^2 = y^2
   => 2^2x + 2^−2x + 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
   => 4^x + 4^−x + 2 = y^2
   By putting the value of 4^x + 4^−x , we get,
   23 + 2= y^2
   y^2 = 25
   Therefore, y = √25 = +5, -5
   The value of x = -5 is rejected since the minimum value of x of 2^x + 2^(-x) is 2 at x=0.
   Using A.M ≥ G.M
   {2^x + 2^(-x)} / 2 ≥ √(2^x* 2^-x)
   2^x + 2^(-x) ≥ 2

Thus statement (2) alone is sufficient to find the value of 2^x + 2^(-x).

Correct Answer: (B)

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