2006 AMC 10A Problem 6

Attempt Problem 6 of the 2006 AMC 10A 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 2006 AMC 10A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

What non-zero real value for

x

satisfies

(7x)^{14} = (14x)^7?

\dfrac{1}{7}

\dfrac{2}{7}

1

7

14

Take the seventh root of both sides

(7x)^2 = 14x,

then divide by

x

Taking the seventh root of both sides gives

(7x)^2 = 14x,

49x^2 = 14x.

Since

x \neq 0,

divide by

x

to get

49x = 14,

hence

x = \dfrac{2}{7}.

Thus, the correct answer is B.

Problem 6 in other years