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

2005 AMC 10B Problem 17 — Solution

2005 AMC 10B Problem 17

Attempt Problem 17 of the 2005 AMC 10B 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 2005 AMC 10B solutions, or check the answer key.

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17.

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

$1$

$\dfrac32$

$2$

$\dfrac52$

$3$

Answer: B

Concepts:exponent telescoping

Difficulty rating: 1480

Solution:

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.

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