14.a) How many numbers between 100 and 200 can be expressed as powers of 2? Show the steps you follow.
b)Which power of 2 is the closest to 1000? Show your work.
I think it is 5, 10x10 or 10^2, 11x11 or 11^2, 12x12 or 12^2, 13x13 or 13^2, 14x14 or 14^2 so total is 5 numbers between 100 to 200 can be expressed as the power of 2. if I read the question correctly.
so i guess we're all grade 7's doing grade 7 mathematics
How many numbers between 100 and 200 can be expressed as a Power of two show the steps you followed
The first part is already answered
2^10 = 1024
so the power of 2 closest to 1000 is 2^10
thanks i understand now!
10 11 12 13 and 14
its pretty easy
a) To determine the numbers between 100 and 200 that can be expressed as powers of 2, follow these steps:
1. Start by finding the smallest power of 2 greater than or equal to 100. From the power of 2 sequence (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...), it is evident that the power of 2 closest to 100 is 2^7 = 128.
2. Next, find the largest power of 2 less than or equal to 200. From the power of 2 sequence, we can see that 2^7 = 128 is less than 200, but 2^8 = 256 is larger. Therefore, the largest power of 2 less than or equal to 200 is 2^7 = 128.
3. Now we have the range of powers of 2 that can be expressed between 100 and 200: 2^7 (128) to 2^7 (128).
4. To calculate the number of powers of 2 between 100 and 200, simply subtract the exponents of the two powers of 2: 7 - 7 = 0.
Therefore, there are 0 numbers between 100 and 200 that can be expressed as powers of 2.
b) To determine which power of 2 is closest to 1000, follow these steps:
1. Begin by examining the power of 2 sequence: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...
2. Look for the first power of 2 that is larger than 1000. From the sequence, we find that 2^10 = 1024 is the first power of 2 greater than 1000.
3. Next, find the last power of 2 that is smaller than 1000. From the sequence, we see that 2^9 = 512 is the last power of 2 below 1000.
4. Compare the two numbers: 1024 and 512. We can see that 1000 falls between these two powers of 2.
5. Calculate the difference between 1000 and the two powers of 2: |1000 - 1024| = 24 and |1000 - 512| = 488.
6. Finally, compare the differences obtained in the previous step. We can see that 24 is smaller than 488.
Therefore, the power of 2 that is closest to 1000 is 2^10 = 1024.