Let's simplify the equation step by step:
Starting with the left side of the equation:
\[ 10^{-3} \times 10 \times 10^{k} \]
We can first combine the exponents with the same base using the property of exponents that states \(a^m \times a^n = a^{m+n}\):
\[ = 10^{-3 + 1 + k} = 10^{-2 + k} \]
Now, we can set this equal to the right side of the equation:
\[ 10^{-2 + k} = 10^{-3} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ -2 + k = -3 \]
Now, solve for \(k\):
\[ k = -3 + 2 \] \[ k = -1 \]
Thus, the value of \(k\) is \(-1\).
So, the corrected interpretation of the equation is:
\[ k = -1 \]