Bunuel wrote:
Is \(a^x\) equal to 4?
(1) \(a^{x + 1} = 4\)
(2) \((a + 1)^x = 4\)
Target question: Does \(a^x = 4\)? Statement 1: \(a^{x + 1} = 4\) Notice that we can rewrite the left side of the equation to get: \((a^{x})(a^{1}) = 4\)
If we ASSUME the answer to the
target question is YES, then that means
\(a^x = 4\)However, we can quickly see that this assumption yields to a contradiction. Here's why:
When we take our newly-expressed equation \((a^{x})(a^{1}) = 4\), and replace \(a^x\) with \(4\), we get: \((4)(a^{1}) = 4\)
Divide both sides of the equation by \(4\) to get: \(a^{1} = 1\), which means it must be the case that \(a = 1\).
HOWEVER, if \(a = 1\),
it's impossible to get \(a^x = 4\)So, we can be certain that
\(a^x\) DOES NOT equal \(4\)Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: \((a + 1)^x = 4\) EDIT: My first response was to raise both sides of the equation by \(\frac{1}{x}\). However, I neglected the fact that, if x is even, doing so would have turned all negative bases into positive values. Thanks to
ThatDudeKnows for pointing me and the right direction!!
So let's try again....
Since \((a + 1)^x = 4\), we can see that \(a = 1\) and \(x = 2 \) is one possible solution. In this case, the answer to the target question is
NO, \(a^x\) DOES NOT equal \(4\)Now, let's see if we can find values of \(a\) and \(x\) that yield a different answer to the target question.
Since statement 2 tells us that \((a + 1)^x = 4\), and since the target question asks,
Does \(a^x = 4\)? , (both equations are set equal to 4), we can
rephrase the target question as:
Does \(a^x = (a + 1)^x \)? Since both sides have the same exponent, we know that \(|a| = |a + 1|\) (as long as x ≠ 0, which it can't)
When we solve \(|a| = |a + 1|\), we get \(a = -\frac{1}{2}\)
When we plug \(a = -\frac{1}{2}\) into the statement 2 equation, we get: \((-\frac{1}{2} + 1)^x = 4\), which means \(x = -2\)
If \(a = -\frac{1}{2}\) and \(x = -2\), then the answer to the target question is
YES, \(a^x\) equals \(4\)Statement 2 is NOT SUFFICIENT
Answer: A