To find the numbers that satisfy \( A \cap C \) (the intersection of sets A and C), we need to identify the numbers that are both perfect squares and perfect fourths.
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Perfect Squares (Set A): The perfect squares from 1 to 100 are:
- \( 1^2 = 1 \)
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)
- \( 6^2 = 36 \)
- \( 7^2 = 49 \)
- \( 8^2 = 64 \)
- \( 9^2 = 81 \)
- \( 10^2 = 100 \)
Thus, the set A is: \[ A = { 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 } \]
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Perfect Fourths (Set C): The perfect fourths from 1 to 100 are:
- \( 1^4 = 1 \)
- \( 2^4 = 16 \)
- \( 3^4 = 81 \) (which is not considered here since \( 3^4 = 81 \) is equal to \( 3^2 = 9 \))
- \( 4^4 = 256 \) (not within 1-100)
Thus, the relevant perfect fourths for our range are: \[ C = { 1, 16 } \]
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Finding the Intersection \( A \cap C \): The intersection \( A \cap C \) consists of the elements that are common in both sets A and C: \[ A \cap C = { 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 } \cap { 1, 16 } \]
This gives us: \[ A \cap C = { 1, 16 } \]
So the numbers that satisfy \( A \cap C \) are: \[ \text{1, 16} \]
The answer is: 1, 16
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