Bunuel wrote:
What does \((1 + m)(1 – m + m^2– m^3) =\) ?
(1) \(m^6 = 64\)
(2) \(m^{-3} = -\frac{1}{8}\)
KEY CONCEPTS: ODD exponents preserve the sign of the base. So, (
NEGATIVE)^(
ODD integer) =
NEGATIVEand (
POSITIVE)^(
ODD integer) =
POSITIVEAn EVEN exponent always yields a positive result (unless the base = 0)
So, (
NEGATIVE)^(
EVEN integer) =
POSITIVEand (
POSITIVE)^(
EVEN integer) =
POSITIVETarget question: What is the value of \((1 + m)(1 – m + m^2– m^3)\)? Statement 1: \(m^6 = 64\) This equation has two possible solutions: \(m = 2\) and \(m = -2\).
STRATEGY: Some students will conclude that, since there are two possible values of m, statement 1 is not sufficient. However, the question does not ask us "What is the value of m?"; it asks us to determine the value of \((1 + m)(1 – m + m^2– m^3)\), and we won't know until we actually test of the two possible values of m. Case a: \(m = 2\) (since \(2^6 = 64\)). In this case, the answer to the target question is
\((1 + m)(1 – m + m^2– m^3) = (1 + 2)(1 – 2 + 2^2– 2^3) = -15\)Case b: \(m = -2\) (since \((-2)^6 = 64\)). In this case, the answer to the target question is
\((1 + m)(1 – m + m^2– m^3) = (1 + (-2))(1 – (-2) + (-2)^2– (-2)^3) = -15\)Aha!!
The two different values of m yield the
same answer to the
target question (the expression evaluates to be -15).
Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: \(m^{-3} = -\frac{1}{8}\)This equation has only one solution \(m = -2\), which means the answer to the target question is
\((1 + m)(1 – m + m^2– m^3) = (1 + (-2))(1 – (-2) + (-2)^2– (-2)^3) = -15\)Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer: D
Cheers,
Brent