Simplify {(256)^{ - (4^{ -3/2})}}

To simplify the expression, let us break it down step by step:

Expression: \( (256)^{-\left(4^{-\frac{3}{2}}\right)} \)

1. Simplify the inner exponent \( 4^{-\frac{3}{2}} \):
- The negative exponent means reciprocal, so:
\( 4^{-\frac{3}{2}} = \frac{1}{4^{\frac{3}{2}}} \).
- Rewrite \( 4^{\frac{3}{2}} \) as a power and root:
\( 4^{\frac{3}{2}} = \left(\sqrt{4}\right)^3 \).
- Since \( \sqrt{4} = 2 \), we have:
\( 4^{\frac{3}{2}} = 2^3 = 8 \).
- Therefore:
\( 4^{-\frac{3}{2}} = \frac{1}{8} \).

2. Substitute \( 4^{-\frac{3}{2}} = \frac{1}{8} \) into the original expression:
\( (256)^{-\left(\frac{1}{8}\right)} \).

3. Simplify \( (256)^{-\frac{1}{8}} \):
- The negative exponent means reciprocal:
\( (256)^{-\frac{1}{8}} = \frac{1}{(256)^{\frac{1}{8}}} \).
- To simplify \( (256)^{\frac{1}{8}} \), recall that \( 256 = 2^8 \):
\( (256)^{\frac{1}{8}} = \left(2^8\right)^{\frac{1}{8}} \).
- Use the property of exponents \( (a^m)^n = a^{m \cdot n} \):
\( \left(2^8\right)^{\frac{1}{8}} = 2^{8 \cdot \frac{1}{8}} = 2^1 = 2 \).
- Therefore:
\( (256)^{\frac{1}{8}} = 2 \).

4. Substitute back into the expression:
\( (256)^{-\frac{1}{8}} = \frac{1}{2} \).

Final Answer: \( \frac{1}{2} \).

Plain Text Answer:
The simplified value of \( (256)^{-\left(4^{-\frac{3}{2}}\right)} \) is \( \frac{1}{2} \).