The sum of all real values of $k$ for which

{\left(\frac{1}{8}\right)}^{\frac{k}{3}}\times {\left(\frac{1}{32768}\right)}^{\frac{1}{3}}=\frac{1}{8}\times {\left(\frac{1}{32768}\right)}^{\frac{k}{1}}

is

The correct option is:
$\frac{-}{\mathrm{2/}}$
which represents the value
$-\frac{2}{3}$.

To understand how this solution is derived, we can simplify the bases of the exponential equation by expressing them with a common base of
$\frac{1}{8}$.

First, let us observe the relationship between the numbers 8 and 32768:
$$8^5=32768$$
Therefore, we can write:
$$\frac{1}{32768}={\left(\frac{1}{8}\right)}^5$$

Substituting this relation into the exponential terms of our equation, the bases match, allowing us to express the exponents on both sides of the equation.
For the term on the left-hand side, we have:
$${\left(\frac{1}{8}\right)}^k\times {\left(\frac{1}{32768}\right)}^{-\frac{1}{3}}={\left(\frac{1}{8}\right)}^k\times {\left[{\left(\frac{1}{8}\right)}^5\right]}^{-\frac{1}{3}}={\left(\frac{1}{8}\right)}^{k-\frac{5}{3}}$$

For the term on the right-hand side, we have:

By equating the exponents of the common base
$\frac{1}{8}$, we obtain the linear equation:
$$k-\frac{5}{3}=1+5k$$

Subtracting
$k$ from both sides gives:
$$-\frac{5}{3}=1+4k$$

Subtracting 1 from both sides:
$$-\frac{5}{3}-1=4k$$
$$-\frac{8}{3}=4k$$

Dividing both sides by 4 yields the solution:
$$k=-\frac{2}{3}$$