2007 AMC 10A Problem 9

Problem 9 of 25EasierAlgebra

Real numbers

a

and

b

satisfy the equations

3^a = 81^{b+2}

and

125^b = 5^{a-3}.

What is

ab?

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Solution

The equations become

3^a = 3^{4(b+2)}

and

5^{3b} = 5^{a-3}.

a = 4(b + 2)

and

3b = a - 3.

Solving gives

a = -12

and

b = -5,

ab = 60.

Thus, the correct answer is E.

Concepts: exponent · system of equations

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.