Step 1: Identify the Expression

We are given the expression $(256)^{0.16} \times (256)^{0.09}$ where the bases are equal.

$(256)^{0.16} \times (256)^{0.09}$

Step 2: Apply Exponent Addition Law

Using the law of exponents $a^m \times a^n = a^{m+n}$, add the decimal exponents: $0.16 + 0.09 = 0.25$.

$(256)^{0.16 + 0.09} = (256)^{0.25}$

Step 3: Convert Decimal Exponent to Fraction

Express $0.25$ as a fraction: $0.25 = \frac{1}{4}$. The expression becomes $(256)^{\frac{1}{4}}$.

$(256)^{\frac{1}{4}}$

Step 4: Simplify Base and Exponent

Since $256 = 4^4$, substitute $256$ with $4^4$ and multiply the powers: $(4^4)^{\frac{1}{4}} = 4^{4 \times \frac{1}{4}} = 4^1 = 4$.

$(4^4)^{\frac{1}{4}} = 4$