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

Key Property: If \(b^x = b^y\), then \(x = y\) as long as \(x \neq 0\), \(x \neq 1\), and \(x \neq -1\).
The provisos (\(x \neq 0\), \(x \neq 1\), and \(x \neq -1\)) play a large role here!

Given: x is a non-zero integer

Target question: What is the value of x?

Statement 1: \(x^{4x} = x^{16}\)
Let's begin by assuming \(x \neq 0\), \(x \neq 1\), and \(x \neq -1\)
Since the bases are equal, the exponents are equal, which means we can write: \(4x = 16\)
So, \(x = 4\) is ONE solution (so far)

We must now test the provisos (other than \(x = 0\), since we're told \(x\) is a nonzero integer)
First let's test \(x = 1\) by plugging it into the equation to get: \(1^{4(1)} = 1^{16}\)
Evaluate to get \(1 = 1\). WORKS!
So, \(x = 1\) is another possible solution.

Now let's test \(x = -1\) by plugging it into the equation to get: \((-1)^{4(-1)} = (-1)^{16}\)
Evaluate to get \(1 = 1\). WORKS!
So, \(x = -1\) is another possible solution.

Since, x can equal 4, 1 or -1, statement 1 is NOT SUFFICIENT


Statement 2: \((x^x)(x^2) = x^6\)
First apply the product law to get: \(x^{x+2} = x^6\)

Once again let's assume \(x \neq 0\), \(x \neq 1\), and \(x \neq -1\)
Since the bases are equal, the exponents are equal, which means we can write: \(x+2 = 6\)
So, \(x = 4\) is ONE solution (so far)

Now let's test \(x = 1\) by plugging it into the equation to get: \(1^{1+2} = 1^6\)
Evaluate to get \(1 = 1\). WORKS!
So, \(x = 1\) is another possible solution.

Now let's test \(x = -1\) by plugging it into the equation to get: \((-1)^{(-1)+2} = (-1)^6\)
Evaluate to get \(-1 = 1\). Doesn't work

Since, x can equal 4 or 1, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x can equal 4, 1 or -1
Statement 2 tells us that x can equal 4 or 1
So when we combine the statements, we know that x can equal 4 or 1

Since we can’t answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

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