How many integers are NOT perfect powers between 1 and 1000?
It might be easier to count how many are powers and subtract that number from 1000. To do this you should first find the highest power of 2 less than 1000, then you'll know how high of powers you have to count. You can check that
29 < 1000 < 210
so you'll need to check up to 9th powers.
For squares find the last square less than 1000 (this is 31^2)
then the last cube less than 1000
Skip fourth powers because they're also squares.
then the last fifth powers less than 1000
Skip sixth powers because they're also squares.
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then the last ninth powers less than 1000
Be sure that you don't double count, because fourth, sixth eigth and tenth powers are also squares, so you really only need to look at prime powers.
You want the first square, cube, fifth, seventh and ninth powers such that they don't exceed 1000, but the next integer does, like 31 and 32 above.
Nine is not prime, so you skip it because the number will be in the cubes too
To find the highest power of 2 less than 1000, determine the exponent that results in a value less than 1000:
2^9 = 512
2^10 = 1024
Since 512 is less than 1000 and 1024 is greater than 1000, the highest power of 2 less than 1000 is 2^9.
Similarly, find the highest power of other prime numbers up to 9.
Square: The last square less than 1000 is 31^2 = 961.
Cube: The last cube less than 1000 is 7^3 = 343.
Fifth power: The last fifth power less than 1000 is 4^5 = 1024.
Seventh power: The last seventh power less than 1000 is 2^7 = 128.
Ninth power: The last ninth power less than 1000 is 2^9 = 512.
Now, count the number of perfect powers between 1 and 1000:
For squares: 31 (since every square number less than or equal to 961 is counted)
For cubes: 7 (since every cube number less than or equal to 343 is counted)
For fifth powers: 4 (since every fifth power number less than or equal to 1024 is counted)
For seventh powers: 2 (since every seventh power number less than or equal to 128 is counted)
For ninth powers: 1 (since every ninth power number less than or equal to 512 is counted)
To find the total number of perfect powers, add up the counts:
31 + 7 + 4 + 2 + 1 = 45
Since there are 45 perfect powers between 1 and 1000, the number of integers that are not perfect powers is:
1000 - 45 = 955
Therefore, there are 955 integers that are not perfect powers between 1 and 1000.
To find the number of integers that are not perfect powers between 1 and 1000, we can use the method described above.
First, let's find the highest power of 2 less than 1000. We can see that 2^9 is less than 1000, but 2^10 is greater. So we will count up to 9th powers.
Next, we need to find the last square less than 1000. To do this, we find the square root of 1000, which is approximately 31.6. The largest square less than 1000 is 31^2.
We then find the last cube less than 1000. The cube root of 1000 is approximately 10, so the largest cube less than 1000 is 10^3.
We skip fourth powers because they are also squares, and we skip sixth powers because they are also squares.
Next, we find the last fifth power less than 1000. The fifth root of 1000 is approximately 4.65. So the largest fifth power less than 1000 is 4^5.
We continue this process, finding the last seventh power less than 1000, the last eighth power less than 1000, and the last ninth power less than 1000.
Finally, we need to make sure that we don't double count. Fourth, sixth, eighth, and tenth powers are also squares, so we only need to look at prime powers.
By following this method, we can count the number of integers that are perfect powers between 1 and 1000. We can then subtract this number from 1000 to get the number of integers that are not perfect powers between 1 and 1000.