AMC10 2006 A

AMC10 2006 A · Q6

AMC10 2006 A · Q6. It mainly tests Linear equations, Exponents & radicals.

What non-zero real value for $x$ satisfies $(7x)^{14} = (14x)^7$?

什么非零实数 $x$ 满足 $(7x)^{14} = (14x)^7$？

(A) $\dfrac{1}{7}$ $\dfrac{1}{7}$

(B) $\dfrac{2}{7}$ $\dfrac{2}{7}$

(D) 7 7

(E) 14 14

Answer

Correct choice: (B)

正确答案：（B）

Solution

(B) Take the seventh root of both sides to get $(7x)^2 = 14x$. Then $49x^2 = 14x$, and because $x \ne 0$ we have $49x = 14$. Thus $x = 2/7$.

（B）对等式两边取七次方根，得到 $(7x)^2 = 14x$。然后 $49x^2 = 14x$，并且因为 $x \ne 0$，所以 $49x = 14$。因此 $x = 2/7$。

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