Question

In the Venn diagram, consider U = {whole numbers 1 – 100}.

Let A represent numbers that are perfect squares, B represent numbers that are perfect cubes, and C represent numbers that are perfect fourths, or numbers with a fourth root that is an integer.



Circles A, B, and C overlap.

Which number(s) satisfy A ∩ C?

16, 81
1, 16, 81
1, 4, 9, 16, 25, 36, 49, 81, 100
1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Answer 1

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.

1. **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:
\[ A = \{ 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 \} \]

2. **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:
\[ C = \{ 1, 16 \} \]

3. **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**