2003 AMC 10B Problem 9

Below is the professionally curated solution for Problem 9 of the 2003 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 10B solutions, or check the answer key.

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Concepts:exponent algebraic manipulation

Difficulty rating: 1070

Find the value of $x$ that satisfies the equation

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

$2$

$3$

$5$

$6$

$9$

Solution:

Writing everything base $5,$ the left side is $5^{-4}$ and the right side is $5^{(48-26-34)/x}=5^{-12/x}.$ Setting exponents equal, $-4=-\dfrac{12}{x},$ so $x=3.$

Thus, the correct answer is B.

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