To solve the expression \( 4^3 \times 4^{-6} \), we can use the properties of exponents. Specifically, we use the property that states:
\[ a^m \times a^n = a^{m+n} \]
In this case, \( a = 4 \), \( m = 3 \), and \( n = -6 \). Therefore, we can combine the exponents:
\[ 4^3 \times 4^{-6} = 4^{3 + (-6)} = 4^{-3} \]
Now, we simplify \( 4^{-3} \):
\[ 4^{-3} = \frac{1}{4^3} \]
Calculating \( 4^3 \):
\[ 4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64 \]
Thus, we have:
\[ 4^{-3} = \frac{1}{64} \]
Finally, the value of the expression \( 4^3 \times 4^{-6} \) is
\[ \boxed{\frac{1}{64}} \]