2016 AMC 10B Problem 1

Problem 1 of 25EasierAlgebra

What is the value of

\dfrac{2a^{-1}+\frac{a^{-1}}{2}}{a}

when

a= \tfrac{1}{2}?

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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 .

Concepts: exponent · substitution

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.