Write each of the following numbers as a power with an exponent to the base 4.
[tex]$\frac{1}{256}$[/tex]
Answer (1)
4 − 4
Explanation
Understanding the Problem We are asked to express the number 256 1 as a power with an exponent to the base 4. This means we want to find a value x such that 4 x = 256 1 .
Expressing 256 as a Power of 4 First, let's express 256 as a power of 4. We can do this by repeatedly multiplying 4 by itself until we reach 256:
4 1 = 4 4 2 = 4 × 4 = 16 4 3 = 16 × 4 = 64 4 4 = 64 × 4 = 256
So, we have 256 = 4 4 .
Using the Negative Exponent Property Now, we can rewrite 256 1 as 4 4 1 . Using the property that a n 1 = a − n , we have 4 4 1 = 4 − 4 .
Final Answer Therefore, 256 1 can be written as 4 − 4 .
Examples
Understanding exponents is crucial in many fields, such as computer science when dealing with memory sizes (e.g., kilobytes, megabytes, gigabytes, which are powers of 2) or in finance when calculating compound interest. For instance, if you invest $1000 at an annual interest rate of 5% compounded quarterly, the amount you'll have after a year can be calculated using exponents. Similarly, expressing numbers as powers of a base helps simplify complex calculations and understand exponential growth or decay in various real-world scenarios.
