Check all that are equivalent to 256.
[tex]$2^8$[/tex]
[tex]$4^4$[/tex]
[tex]$16^2$[/tex]
[tex]$128^2$[/tex]
Answer (1)
Calculate 2 8 which equals 256.
Calculate 4 4 which equals 256.
Calculate 1 6 2 which equals 256.
Calculate 12 8 2 which equals 16384.
Therefore, 2 8 , 4 4 , and 1 6 2 are equivalent to 256. 2 8 , 4 4 , 1 6 2
Explanation
Understanding the Problem We are given the number 256 and asked to determine which of the following expressions are equivalent to it: 2 8 , 4 4 , 1 6 2 , 12 8 2 .
Plan of Action We will evaluate each expression and compare the result to 256.
Evaluating 2 8 Let's start with the first expression, 2 8 . This means 2 multiplied by itself 8 times: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 . The result of this calculation is 256.
Evaluating 4 4 Next, we evaluate 4 4 . This means 4 multiplied by itself 4 times: 4 × 4 × 4 × 4 . The result of this calculation is 256.
Evaluating 1 6 2 Now, we evaluate 1 6 2 . This means 16 multiplied by itself 2 times: 16 × 16 . The result of this calculation is 256.
Evaluating 12 8 2 Finally, we evaluate 12 8 2 . This means 128 multiplied by itself 2 times: 128 × 128 . The result of this calculation is 16384.
Conclusion Comparing the results to 256, we find that 2 8 = 256 , 4 4 = 256 , and 1 6 2 = 256 , while 12 8 2 = 16384 . Therefore, 2 8 , 4 4 , and 1 6 2 are equivalent to 256.
Examples
Understanding exponents and their equivalent forms is crucial in various fields, such as computer science and engineering. For instance, in computer memory, sizes are often expressed as powers of 2 (e.g., 256MB, 512MB, 1024MB which is 1GB). Knowing that 2 8 equals 256 helps in quickly understanding memory capacities or data storage sizes. Similarly, in image processing, image dimensions or pixel counts can be related to powers of numbers, making it easier to analyze and manipulate images efficiently.
