An integer is randomly chosen from the integers 1 through 100, inclusive. What is the probability that the chosen integer is a perfect square or a perfect cube, but not both? Express your answer as a common fraction.
squares ... 1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100
cubes ... 1 , 8 , 27 , 64
Why did the number go to therapy? Because it had too many square and cube issues! 🤡
Now, let's solve this problem together. We need to find the probability of choosing an integer that is either a perfect square or a perfect cube, but not both.
First, let's find the total number of integers from 1 to 100. There are 100 integers in that range.
To find the number of perfect squares, we take the square root of 100, which is 10. So, there are 10 perfect squares from 1 to 100.
Next, let's find the number of perfect cubes. The cube root of 100 is approximately 4.64. Since we can only choose integers, there are 4 perfect cubes from 1 to 100.
Now, let's calculate the number of integers that are both perfect squares and perfect cubes. Since the only overlapping number is 1, there is just one such number.
So, the total number of integers that are either perfect squares or perfect cubes, but not both, is (10 + 4) - 1 = 13.
Finally, to find the probability, we divide the number of desired outcomes (13) by the total number of possible outcomes (100):
13 / 100 = 13/100.
Therefore, the probability of choosing an integer that is a perfect square or a perfect cube, but not both, is 13/100.
To find the probability that the chosen integer is a perfect square or a perfect cube but not both, we need to calculate the following:
Probability of selecting a perfect square but not a perfect cube + Probability of selecting a perfect cube but not a perfect square.
Step 1: Calculate the number of perfect squares from 1 to 100.
The perfect squares from 1 to 100 are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. So, there are a total of 10 perfect squares.
Step 2: Calculate the number of perfect cubes from 1 to 100.
The perfect cubes from 1 to 100 are 1, 8, 27, 64. So, there are a total of 4 perfect cubes.
Step 3: Calculate the number of integers that are both perfect squares and perfect cubes.
The only integer that is both a perfect square and a perfect cube is 1.
Step 4: Calculate the probability of selecting a perfect square but not a perfect cube.
The probability of selecting a perfect square but not a perfect cube is (10 - 1)/100 = 9/100.
Step 5: Calculate the probability of selecting a perfect cube but not a perfect square.
The probability of selecting a perfect cube but not a perfect square is (4 - 1)/100 = 3/100.
Step 6: Calculate the total probability.
Finally, to find the probability of selecting a perfect square or a perfect cube but not both, we add the probabilities from steps 4 and 5:
9/100 + 3/100 = 12/100 = 3/25.
Therefore, the probability that the chosen integer is a perfect square or a perfect cube, but not both, is 3/25.
To find the probability that the chosen integer is a perfect square or a perfect cube, but not both, we need to determine the number of integers that meet this condition.
First, let's find the number of perfect squares from 1 to 100. The perfect squares are the numbers that can be expressed as the square of an integer. So, we need to find the number of integers whose square is between 1 and 100. The square of an integer can be at most 10^2 (since 11^2 = 121). Therefore, there are 10 perfect squares from 1 to 100.
Next, let's find the number of perfect cubes from 1 to 100. The perfect cubes are the numbers that can be expressed as the cube of an integer. So, we need to find the number of integers whose cube is between 1 and 100. The cube of an integer can be at most 4^3 (since 5^3 = 125). Therefore, there are 4 perfect cubes from 1 to 100.
Now, we need to find the number of integers that are both perfect squares and perfect cubes. These numbers are the ones that can be expressed as both the square and the cube of an integer. The possible values are the numbers that are both perfect squares and perfect cubes between 1 and 100. Squaring an integer from 1 to 4 results in a perfect square, which is also a perfect cube. Therefore, there are 4 integers that are both perfect squares and perfect cubes.
To find the number of integers that are either a perfect square or a perfect cube, but not both, we can sum the number of perfect squares (10) and the number of perfect cubes (4), and then subtract the number of integers that are both (4).
10 + 4 - 4 = 10
So, there are 10 integers from 1 to 100 that are either a perfect square or a perfect cube, but not both.
Since there are 100 integers from 1 to 100, the probability of randomly selecting one of these integers is 10/100, which simplifies to 1/10.
Therefore, the probability is 1/10.