2003 AMC 10B Problem 9

Problem 9 of 25EasierAlgebra

Find the value of

x

that satisfies the equation

25^{-2} = \dfrac{5^{\frac{48}{x}}}{5^{\frac{26}{x}} \cdot 25^{\frac{17}{x}}}.

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Solution

Writing everything base

5,

the left side is

5^{-4}

and the right side is

5^{\frac{48-26-34}{x}}=5^{-\frac{12}{x}}.

Setting exponents equal,

-4=-\dfrac{12}{x},

x=3.

Thus, the correct answer is B.

Concepts: exponent · algebraic manipulation

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