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

### Concept & Formula When multiplying expressions with the same base, simply add their exponents according to the fundamental laws of indices. $$ a^m \times a^n = a^{m+n} $$ ### Step-by-Step Solution * Given the expression: $(256)^{0.16} \times (256)^{0.09}$ * Since the bases ($256$) are identical, we apply the multiplication rule and add the exponents. * $0.16 + 0.09 = 0.25$ * The expression simplifies to: $(256)^{0.25}$ * Convert the decimal exponent to a simple fraction for easier calculation: $0.25 = \frac{1}{4}$. * Now evaluate $(256)^{\frac{1}{4}}$. We need to find a number which, when multiplied by itself four times, equals $256$. * Since $4^4 = 256$, we can rewrite the expression as $(4^4)^{\frac{1}{4}}$. * Multiply the exponents: $4 \times \frac{1}{4} = 1$. * The result is $4^1 = 4$. ### Exam Strategy & Shortcut If you memorize powers of basic numbers (like $4^4 = 256$ or $2^8 = 256$), you can skip intermediate steps. Seeing the sum of exponents is $0.25$ (a quarter), immediately find the 4th root of $256$. Knowing $4 \times 4 \times 4 \times 4 = 256$ takes you straight to the answer. ### Common Pitfall Students sometimes multiply the bases together or multiply the exponents ($0.16 \times 0.09$) instead of adding them when dealing with multiplication of identical bases. Always stick to the addition rule for exponents with the same base. ### Final Answer **Therefore, the correct answer is 4.**