2004 AMC 10B Problem 5

In the expression

c \cdot a^b - d,

the values of

a,

b,

c,

and

d

are

0,

1,

2,

and

3,

although not necessarily in that order. What is the maximum possible value of the result?

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Solution

Setting

d = 0

removes the subtraction, so we maximize

c \cdot a^b

using

1,

2,

3.

Taking

c = 1,

a = 3,

b = 2

gives

3^2 = 9.

The alternative

2^3 = 8

is smaller, and any assignment with

c \gt 1

forces a smaller power. The maximum is

9.

Thus, the correct answer is D.

Tagged: exponent · optimization · casework