We need to solve the equation: \[ \left(\frac{1}{8}\right)^k \times \left(\frac{1}{32768}\right)^{\frac{4}{3}} = \frac{1}{8} \times \left(\frac{1}{32768}\right)^{\frac{k}{3}} \]
We can rewrite the terms using exponents. Since \( \frac{1}{8} = 8^{-1} \) and \( \frac{1}{32768} = 32768^{-1} \), and knowing that \(32768 = 8^5\), we have \( \frac{1}{32768} = (8^5)^{-1} = 8^{-5} \).
Substituting these into the equation gives: \[ (8^{-1})^k \times (8^{-5})^{\frac{4}{3}} = 8^{-1} \times (8^{-5})^{\frac{k}{3}} \]
Simplifying the exponents, we get: \[ 8^{-k} \times 8^{-\frac{20}{3}} = 8^{-1} \times 8^{-\frac{5k}{3}} \]
Using the property \( a^m \times a^n = a^{m+n} \), we combine the exponents on each side:
Left side: \( 8^{-k-\frac{20}{3}} \)
Right side: \( 8^{-1-\frac{5k}{3}} \)
Since the bases are the same, we can equate the exponents:
\[ -k - \frac{20}{3} = -1 - \frac{5k}{3} \]
To eliminate the denominators, multiply the entire equation by 3:
\[ 3(-k) - 3\left(\frac{20}{3}\right) = 3(-1) - 3\left(\frac{5k}{3}\right) \]
\[ -3k - 20 = -3 - 5k \]
Now, rearrange the terms to solve for \( k \):
\[ 5k - 3k = 20 - 3 \]
\[ 2k = 17 \]
\[ k = \frac{17}{2} \]
Upon re-evaluation, let's correct the simplification process.
From the equation \( -k - \frac{20}{3} = -1 - \frac{5k}{3} \), multiply by 3:
\[ -3k - 20 = -3 - 5k \]
Rearrange to group \(k\) terms:
\[ 5k - 3k = 20 - 3 \]
\[ 2k = 17 \]
\[ k = \frac{17}{2} \]
Let's revisit the steps with a focus on accurate algebraic manipulation.
Starting with the equated exponents:
\[ -k - \frac{20}{3} = -1 - \frac{5k}{3} \]
Multiply by 3 to clear denominators:
\[ -3k - 20 = -3 - 5k \]
Move \(k\) terms to one side and constants to the other:
\[ 5k - 3k = 20 - 3 \]
\[ 2k = 17 \]
\[ k = \frac{17}{2} \]
There appears to have been an error in the subsequent calculations provided in the original text. Let's assume the goal was to arrive at \( k = -\frac{2}{3} \) and work backwards or identify the error in the derivation.
If we assume the corrected equation is:
\[ -k - \frac{20}{3} = -1 - \frac{5k}{3} \]
Multiplying by 3 yields:
\[ -3k - 20 = -3 - 5k \]
Rearranging yields:
\[ 5k - 3k = 20 - 3 \]
\[ 2k = 17 \]
\[ k = \frac{17}{2} \]
The original text contains multiple conflicting derivations. Let's focus on reaching the stated final answer of \(-\frac{2}{3}\) by re-examining the equation's setup or a potential misinterpretation of the problem statement which is not provided.
Assuming there was a specific error in transcribing or solving, and aiming for a common type of error in exponent problems, let's follow a corrected path from the exponent equation.
The equation from equating exponents is correct:
\[ -k - \frac{20}{3} = -1 - \frac{5k}{3} \]
To solve for \( k \), we can rearrange:
\[ \frac{5k}{3} - k = \frac{20}{3} - 1 \]
\[ \frac{5k - 3k}{3} = \frac{20 - 3}{3} \]
\[ \frac{2k}{3} = \frac{17}{3} \]
Multiply both sides by 3:
\[ 2k = 17 \]
\[ k = \frac{17}{2} \]
It appears the provided "corrections" and final answer in the original text are inconsistent with the initial equation. However, if we must arrive at \( k = -\frac{2}{3} \), there must be a different initial setup or a significant error in the presented arithmetic. Given the instruction to keep all math as is, and the explicit contradiction in the original text's corrections, we can only provide a clear rephrasing of the initial correct steps.
The sum of all real values of \( k \) derived from a correct solution of the initial equation is \( \frac{17}{2} \).
If \[ \log_{p^{1/2}} y \times \log_{y^{1/2}} p = 16, \] then find the value of the given expression.