AMC10 2005 B

AMC10 2005 B · Q17

AMC10 2005 B · Q17. It mainly tests Exponents & radicals.

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

假设 $4^a = 5$，$5^b = 6$，$6^c = 7$，且 $7^d = 8$。$a \cdot b \cdot c \cdot d$ 是多少？

(A) 1 1

(B) $\frac{3}{2}$ $\frac{3}{2}$

(D) $\frac{5}{2}$ $\frac{5}{2}$

(E) 3 3

Answer

Correct choice: (B)

正确答案：（B）

Solution

(B) Because $4^{a\cdot b\cdot c\cdot d}=\left(\left(\left(4^a\right)^b\right)^c\right)^d=\left(\left(5^b\right)^c\right)^d=\left(6^c\right)^d=7^d=8=4^{3/2},$ we have $a\cdot b\cdot c\cdot d=3/2.$

（B）因为 $4^{a\cdot b\cdot c\cdot d}=\left(\left(\left(4^a\right)^b\right)^c\right)^d=\left(\left(5^b\right)^c\right)^d=\left(6^c\right)^d=7^d=8=4^{3/2},$ 所以有 $a\cdot b\cdot c\cdot d=3/2.$

Topics

AL.008 Exponents & radicals