Question

设 $4^a = 5$ 、 $5^b = 6$ 、 $6^c = 7$ ，且 $7^d = 8$ 。求 $a \cdot b \cdot c \cdot d$ 。

Suppose that $4^a = 5,$ $5^b = 6,$ $6^c = 7,$ and $7^d = 8.$ What is $a \cdot b \cdot c \cdot d?$

$1$

$\dfrac32$

$2$

$\dfrac52$

$3$

答案：B

知识点：指数裂项相消

难度评级：1480

解答：

将方程串联起来： $\begin{aligned} 4^{abcd} &= \left(\left(\left(4^a\right)^b\right)^c\right)^d \\ &= \left(\left(5^b\right)^c\right)^d \\ &= \left(6^c\right)^d = 7^d = 8. \end{aligned}$

因为 $8 = 4^{3/2}$ ，所以 $a \cdot b \cdot c \cdot d = \dfrac32$ 。

所以正确答案是 B。

Chaining the equations, $\begin{aligned} 4^{abcd} &= \left(\left(\left(4^a\right)^b\right)^c\right)^d \\ &= \left(\left(5^b\right)^c\right)^d \\ &= \left(6^c\right)^d = 7^d = 8. \end{aligned}$

Since $8 = 4^{3/2},$ we conclude $a \cdot b \cdot c \cdot d = \dfrac32.$

Thus, B is the correct answer.

2005 AMC 10B 第 17 题

答案：B

难度评级：1480

解答：

其他年份的第 17 题