Manhattan challenge problem of the week - June 19

If x is a non-zero integer, what is the value of x ^ y?

(1) x = 2
(2) (128 ^ x)[6 ^ (x + y)] = (48 ^ 2x)(3 ^ -x)

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Answer : must be B

Statement (1) - not sufficient - value of y is still not known.

Statement (2)- Sufficient.

(128 ^ x)(6 ^ x)(6 ^ y) = [(48 ^x) (48 ^ x)] / (3 ^ x)

=> (16.8 ^ x)(6 ^ x)(6 ^ y) = (16 ^ x) (48 ^ x)

=> (16 ^ x)(8 ^ x)(6 ^ x)(6 ^ y) = (16 ^ x) (8 ^ x)(6 ^ x)
=> (6 ^ y) = 1
=> y = 0
=> (x^y) = 1 , because any number raised to the power zero is equal to 1.

Hence B must be the answer.

Official Answer and Explanation to the above problem

One of the most effective ways to begin solving problems involving exponential equations is to break down bases of the exponents into prime factors and combine exponents with the same base. Following this approach, be sure to simplify each statement as much as possible before arriving at the conclusion, since difficult problems with exponents often result in unobvious outcomes.

(1) INSUFFICIENT: While this statement gives us the value of x, we know nothing about y and cannot determine the value of x^y.

(2) SUFFICIENT: (128^x)[6^(x + y)] = (48^2x)(3^-x)

[(2^7)^x][(2 × 3)^(x + y)] = {(2^4 ) 3]^2x}(3^-x)

[(2^7)^x][2^(x + y)][3^(x + y)] = (2^8x)(3^2x)(3^-x)

[2^(8x + y)][3^(x + y)] = (2^8x)[3^(2x - x)]

(2^8x)( 2^y)(3^x)(3^y) = (2^8x)(3^x)

( 2^y)(3^y) = 1

(2 × 3)^y = 1

6^y = 1

y = 0

Since y = 0 and x is not equal to zero (as stated in the problem stem), this information is sufficient to conclude that x^y = x^0 = 1.

The correct answer is B.