Find the value of x that satisfies the equation 25^-2 = \dfrac5^(48)/(x)5^(26)/(x) × 25^(17)/(x).

2003 AMC 10B Problem 9 — Solution

2003 AMC 10B Problem 9

Attempt Problem 9 of the 2003 AMC 10B below, then check your answer against the professionally curated solution 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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Find the value of $x$ that satisfies the equation

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

$2$

$3$

$5$

$6$

$9$

Answer: B

Concepts:exponent algebraic manipulation

Difficulty rating: 1070

Small Hint:

Rewrite every term as a power of $5$

Big Hint:

The right side becomes $5^{\frac{48-26-34}{x}};$ set the exponent equal to $-4$

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, $-4=-\dfrac{12}{x},$ so $x=3.$

Thus, the correct answer is B.

Problem 8#8

Full Exam

Problem 10#10

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