Let's solve the expression step by step. The expression is:
\[ \frac{2}{3} \cdot 6^{-(6-4)^3} \]
First, simplify the exponent in the expression:
\[ 6 - 4 = 2 \]
Now substituting this into the exponent:
\[ (6 - 4)^3 = 2^3 = 8 \]
Next, substitute this back into the expression:
\[ \frac{2}{3} \cdot 6^{-8} \]
This can be rewritten as:
\[ \frac{2}{3} \cdot \frac{1}{6^8} \]
Now, multiply the fractions together:
\[ \frac{2}{3 \cdot 6^8} \]
6 can be factored as \(2 \cdot 3\), so:
\[ 6^8 = (2 \cdot 3)^8 = 2^8 \cdot 3^8 \]
This means:
\[ \frac{2}{3 \cdot 6^8} = \frac{2}{3 \cdot (2^8 \cdot 3^8)} = \frac{2}{2^8 \cdot 3^9} \]
Now simplifying this gives:
\[ \frac{1}{2^7 \cdot 3^8} \]
Thus, the final answer for the expression is:
\[ \frac{1}{2^7 \cdot 3^8} \]
or approximately:
\[ \frac{1}{128 \cdot 6561} \]
If you calculate \(128 \cdot 6561\), you get 8388608, leading us to:
\[ \frac{1}{8388608} \]
This is the simplified value of the given expression.