To find the numerical equivalent of \( 81 \cdot 83 \), we can use the difference of squares formula:
\[ a^2 - b^2 = (a-b)(a+b) \]
In this case, we can set \( a = 82 \) (the average of 81 and 83) and \( b = 1 \) (the offset from the average):
\[ 81 \cdot 83 = (82 - 1)(82 + 1) = 82^2 - 1^2 \]
Calculating \( 82^2 \):
\[ 82^2 = 6724 \] \[ 1^2 = 1 \]
Now, substituting back:
\[ 81 \cdot 83 = 6724 - 1 = 6723 \]
Thus, none of the provided responses (512, \( 8^4 \) which equals 4096, or 520) is correct for the product \( 81 \cdot 83 \). The correct calculation shows:
\[ 81 \cdot 83 = 6723 \]