The value of $(256)^{\frac{5}{4}}$ is

### Concept & Formula This problem evaluates your ability to work with fractional exponents. The denominator of a fractional exponent represents the root, and the numerator represents the power. Finding a prime base is the most efficient way to solve this. $$ x^{\frac{m}{n}} = (\sqrt[n]{x})^m = (x^{\frac{1}{n}})^m $$ ### Step-by-Step Solution * **Given:** The expression to evaluate is $(256)^{\frac{5}{4}}$. * **Calculation:** 1. Identify the base ($256$) and express it as a smaller exponent to easily interact with the fractional power. We know $256 = 4^4$. (Alternatively, $2^8$). Let us use $4^4$ as it perfectly cancels the denominator. 2. Substitute this into the expression: $(4^4)^{\frac{5}{4}}$ 3. Apply the power of a power rule $(x^a)^b = x^{ab}$ by multiplying the exponents: $4 \times \frac{5}{4} = 5$ 4. The expression simplifies to: $4^5$ 5. Calculate the final power: $4^5 = 4^4 \times 4 = 256 \times 4 = 1024$ ### Exam Strategy & Shortcut Look at the denominator of the fractional exponent—it is $4$. This is a major clue that $256$ is something to the power of $4$. Recognizing $256$ as $4^4$ instantly allows you to cancel the $4$s, leaving you with $4^5$. If you know your powers of $2$, $2^{10}$ is a famous benchmark number in computer science (1 kilobyte), which is $1024$. ### Common Pitfall A standard pitfall is trying to calculate the $4$th root of $256$ without converting it to a base exponent first. While finding $\sqrt[4]{256} = 4$ is doable, students often get stuck on larger numbers if they do not instinctively break them down into prime or smaller bases. ### Final Answer **Therefore, the correct answer is 1024.**