Question

nd venn diagrams
in the venn diagram, let u be the set of all whole numbers from 1 to 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.
which number(s) satisfy a ∩ c?
16, 81
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
1, 4, 9, 16, 25, 36, 49, 81, 100
1, 16, 81

Explanation:

Step1: Define \( A \cap C \)

\( A \cap C \) is the set of numbers that are in both \( A \) (perfect squares) and \( C \) (perfect fourths). A perfect fourth is also a perfect square (since \( n^4=(n^2)^2 \)), so we need numbers that are perfect fourths (or have integer fourth roots) and perfect squares.

Step2: Find perfect fourths in 1 - 100

Perfect fourths: \( 1^4 = 1 \), \( 2^4 = 16 \), \( 3^4 = 81 \), \( 4^4 = 256 \) (too big). So perfect fourths in 1 - 100 are 1, 16, 81. These are also perfect squares (\( 1 = 1^2 \), \( 16 = 4^2 \), \( 81 = 9^2 \)).

Step3: Match with options

Check options:

• First option: 16, 81 (missing 1)
• Second option: All perfect squares (includes non - fourths like 9, 25, etc.)
• Third option: Same as second (includes non - fourths)
• Fourth option: 1, 16, 81 (matches our result)

Answer:

1, 16, 81 (the fourth option: 1, 16, 81)