AMC 10 Daily Practice Round 1

Complete problem set with solutions and individual problem pages

Problem 13 Medium

Real numbers x and y satisfy {{2}^{3x-5y}}=2048 and {{3}^{x-2y}}=27. What is the value of {{4}^{x-y}}?

  • A.

    128

  • B.

    256

  • C.

    512

  • D.

    1024

  • E.

    2048

Answer:D

\begin{eqnarray} {{2}^{3x-5y}}&=&2048\\ {{2}^{3x-5y}}&=&{{2}^{11}}\\ 3x-5y&=&11, \end{eqnarray}

\begin{eqnarray}{{3}^{x-2y}}&=&27\\ {{3}^{x-2y}}&=&{{3}^{3}}\\ x-2y&=&3，\end{eqnarray}

then, x=7, y=2.

Thus, {{4}^{x-y}}={{4}^{7-2}}={{4}^{5}}=1024.

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