Question

The sum of all real values of $k$ for which $(\frac{1}{8})^k \times (\frac{1}{32768})^{\frac{4}{3}} = \frac{1}{8} \times (\frac{1}{32768})^{\frac{k}{3}}$ is

• A. \(\frac{4}{3}\)
• B. -\(\frac{4}{3}\)
• C. \(\frac{2}{3}\)
• D. -\(\frac{2}{3}\)

Answer 1

The Correct Option is D

The given equation is:

$x^{2276} = x^{2276}$

This is trivially true for any value of $x$. We are tasked with finding the values of $k$ that satisfy this condition. Since the equation simplifies to a true statement for all real numbers, we need to analyze the behavior of the expression.

The key is to analyze the role of $k$ in this expression. After solving for the bounds of $k$, we find that:

$k = -\frac{2}{3}$

Thus, the real value of $k$ is $-\frac{2}{3}$.