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

2016 AMC 10B Problem 1

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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

Substitute

a=\frac{1}{2}

after combining powers of

a

Dividing by

a

turns

a^{-1}

into

a^{-2}

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 .