Prove a (xy) = a x + a y and simplify 4 2 + 4 32 - 4 0.25.

Given $\log_a x = p$, we can write this as $x = a^p$.
Given $\log_a y = q$, we can write this as $y = a^q$.

Using the exponent rule $a^m \cdot a^n = a^{m+n}$:

This shows the first part.

Take the logarithm with base $a$ on both sides of the equation $xy = a^{p+q}$:

Using the logarithm property $\log_b (b^M) = M$:

Substitute back the original definitions of $p$ and $q$:

This deduces the required property.

a) (ii)

Combine the first two terms:

Combine the remaining terms:

So,

We need to find the power to which 4 must be raised to get 256.
$4^1 = 4$
$4^2 = 16$
$4^3 = 64$
$4^4 = 256$
Therefore, $\log_4 256 = 4$.

The simplified value is $\boxed{\text{4}}$.