Question

Ted 错把写成了 $2^m\cdot\sqrt{\dfrac{1}{4096}}$ $2\cdot\sqrt[m]{\dfrac{1}{4096}}.$

求使这两个表达式值相等的所有实数 $m$ 之和。

Ted mistakenly wrote $2^m\cdot\sqrt{\dfrac{1}{4096}}$ as $2\cdot\sqrt[m]{\dfrac{1}{4096}}.$

What is the sum of all real numbers $m$ for which these two expressions have the same value?

$5$

$6$

$7$

$8$

$9$

答案：C

知识点：指数二次方程

难度评级：1280

解答：

可将 $4096$ 写成 $2^{12}$ ，所以 $\dfrac{1}{4096} = 2^{-12}$ 。令两个给定表达式相等，得到比较指数得 $2^m \cdot 2^{-6} = 2 \cdot 2^{\frac{-12}{m}}.$ $m - 6 = 1 + \dfrac{-12}{m}.$

两边乘以 $m$ ，得到因而 $m^2 - 6m = m - 12$ $m^2 - 7m + 12 = 0$ $(m-4)(m-3)$ $m=4,~m=3$

因此，所有解的和为 $7$ 。

所以正确答案是 C。

We can rewrite $4096$ as $2^{12},$ so $\dfrac{1}{4096} = 2^{-12}.$ Then if we equate the given expressions, we get $2^m \cdot 2^{-6} = 2 \cdot 2^{\frac{-12}{m}}.$ Equating the exponents, we get $m - 6 = 1 + \dfrac{-12}{m}.$

Multiplying by $m,$ we get $m^2 - 6m = m - 12$ and so $m^2 - 7m + 12 = 0$ $(m-4)(m-3)$ $m=4,~m=3$

Therefore, we can see that the sum of the solutions is $7.$

Thus, C is the correct answer.

2022 AMC 10A 第 11 题

答案：C

难度评级：1280

解答：

其他年份的第 11 题