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Exponents Equation

XAT 2013 · Quantitative Ability & Data Interpretation · Medium · Algebra

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p and q are positive numbers such that pq=qpp^q = q^p, and q=9pq = 9p. The value of p is

  1. A.

    9\sqrt{9}

  2. B.

    96\sqrt[6]{9}

  3. C.

    99\sqrt[9]{9}

  4. D.

    98\sqrt[8]{9}

  5. E.

    93\sqrt[3]{9}

Answer

D

Explanation

Substitute q=9pq = 9p into pq=qpp^q = q^p: p9p=(9p)p  ⟹  (p9)p=(9p)p  ⟹  p9=9p  ⟹  p8=9  ⟹  p=91/8=98p^{9p} = (9p)^p \implies (p^9)^p = (9p)^p \implies p^9 = 9p \implies p^8 = 9 \implies p = 9^{1/8} = \sqrt[8]{9}.

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