What is the value of \dfrac2a^-1+\fraca^-12a when a= (1)/(2)?

2016 AMC 10B Problem 1 — Solution

2016 AMC 10B Problem 1

Attempt Problem 1 of the 2016 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 2016 AMC 10B solutions, or check the answer key.

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What is the value of $\dfrac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \tfrac{1}{2}?$

$\ 1$

$\ 2$

$\ \dfrac{5}{2}$

$\ 10$

$\ 20$

Answer: D

Concepts:exponent substitution

Difficulty rating: 770

Small Hint:

Substitute $a=\frac{1}{2}$ after combining powers of $a$

Big Hint:

Dividing by $a$ turns $a^{-1}$ into $a^{-2}$

Solution:

The expression is equivalent to $2a^{-2}+\frac{a^{-2}}{2} = 2.5 (a^{-1})^2.$ Then $a^{-1}$ is equal to $\dfrac{1}{\frac 12} = 2,$ so our expression is equal to $2.5\cdot 2^2 = 10.$

Thus, the correct answer is D .

Full Exam

Problem 2#2

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