Question

the digits 2, 0, 2, and 4 are placed in the expression below, one digit per box. what is the minimum possible value of the expression? (2023 amc 8 problems, question #6)

\\ \square^{\square} - \square^{\square} \\

a 4
b 0
c -2
d -4
e -16

Explanation:

Define the expression and digits

The expression is of the form:

$$ A^B - C^D $$

where \(\{A, B, C, D\}\) is a permutation of the digits \(\{2, 0, 2, 4\}\).

Minimize the expression

To minimize \(A^B - C^D\), we want to make \(A^B\) as small as possible and \(C^D\) as large as possible.

• The largest possible value for \(C^D\) using digits from \(\{2, 0, 2, 4\}\) is \(2^4 = 16\) or \(4^2 = 16\).
• If we set \(C^D = 2^4 = 16\), the remaining digits for \(A^B\) are \(\{0, 2\}\).
• The possible values for \(A^B\) using \(\{0, 2\}\) are:
• \(0^2 = 0\)
• \(2^0 = 1\)
• Choosing \(A^B = 0^2 = 0\) gives:

$$ 0^2 - 2^4 = 0 - 16 = -16 $$

Verify other combinations

• If \(C^D = 4^2 = 16\), the remaining digits for \(A^B\) are \(\{0, 2\}\), yielding \(0^2 - 4^2 = -16\).
• Any other combination yields a larger value (e.g., \(2^0 - 2^4 = 1 - 16 = -15\), or positive values).
• Thus, the minimum possible value is \(-16\).

Answer:

• A 4
• B 0
• C -2
• D -4
• E -16 (Correct answer)