how many integral powers of 2 are there between 2 and 2000?
2,4,8,16,256,512,1024,2048,4096, ...
Well, it's a good thing that I'm not afraid of numbers, unlike clowns and dentists! So, let's count the integral powers of 2 between 2 and 2000.
Starting from 2, we have 2^1 = 2, then 2^2 = 4, 2^3 = 8, and so on. If we continue this pattern, we realize that 2 raised to the power of any positive integer will be greater than the previous one.
Now, we need to find the highest power of 2 less than 2000. By trial and error, we can determine that 2^10 = 1024 is the highest power of 2 less than 2000.
So, we have counted 10 integral powers of 2 between 2 and 2000. Don't worry, they won't put on a magic show or try to sell you balloon animals.
To find the number of integral powers of 2 between 2 and 2000, we need to determine the highest power of 2 that is less than or equal to 2000.
Since 2 is the first power of 2, we can start with 2^1 = 2. We keep multiplying by 2 until we reach a value greater than 2000.
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512
2^10 = 1024
2^11 = 2048
As we can see, 2^11 (which is 2048) is the highest power of 2 less than 2000.
Therefore, there are 11 integral powers of 2 (from 2^1 to 2^11) between 2 and 2000.
To find the number of integral powers of 2 between 2 and 2000, we need to identify the range of powers of 2 that fall within this interval.
In this case, the lowest power of 2 within the interval is 2^1 (which is equal to 2) and the highest power of 2 within the interval is 2^10 (which is equal to 1024). We disregard 2^0 since it represents 1, which is not greater than 2.
Therefore, there are a total of 10 powers of 2 between 2 and 2000.