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If $a^{1/a}=b^{1/b}=c^{1/c}$ and $a^{bc}+b^{ac}+c^{ab}=729$, which of the following equals to $a^{1/a}$?

  1. $\sqrt[abc]{81}$
  2. $\sqrt{2}$
  3. $\sqrt[abc]{27}$
  4. $\sqrt[abc]{9}$

This question is from the book, Mathematics, Class 9 (The IIT Foundation Series) , page number 1.25, question number 58.

My attempts to solve this question have failed several times. However, I did find the value of $a^{1/a}$ but not in the correct format. Below is my method to do so.

$$a^{1/a}=b^{1/b}=c^{1/c}$$ $$\Rightarrow \sqrt[a]{a}=\sqrt[b]{b}=\sqrt[c]{c}$$ $$\Rightarrow (\sqrt[a]{a})^{abc}=(\sqrt[b]{b})^{abc}=(\sqrt[c]{c})^{abc}$$ $$\Rightarrow a^{bc}=b^{ac}=c^{ab}$$

Here I conclude our first equation, $a^{bc}=b^{ac}=c^{ab}$. Moving on to the next equation, we have:

$$a^{bc}+b^{ac}+c^{ab}=729$$ $$\Rightarrow a^{bc}+b^{ac}+c^{ab}=729$$ $$\Rightarrow a^{bc}+a^{bc}+a^{bc}=729$$ $$\Rightarrow 3a^{bc}=729$$ $$\Rightarrow a^{bc}=243$$ $$\Rightarrow (a^{bc})^{1/abc}=243^{1/abc}$$ $$\Rightarrow a^{1/a}=\sqrt[abc]{243}$$ $$\Rightarrow a^{1/a}=\sqrt[abc]{3^5}$$

Here I finally find the value of $a^{1/a}$ as $\sqrt[abc]{3^5}$. However, none of the options match with my result. Please help me to solve the question completely. Thanks!

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    $\begingroup$ You've shown that none of the options can be correct. $\endgroup$
    – Toby Mak
    May 4 at 6:09
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    $\begingroup$ @TobyMak If that’s the case, thanks for notifying. I had been smashing my head for an incorrect question. Also, thanks for your time on this question. Your presence was really helpful! $\endgroup$ May 4 at 6:14
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I think the question is wrong.

Your math is correct that means that none of the options are right and since that is not in the options, the question is incorrect.

By the way if you investigate then you will notice that the question would work out if

$a^{bc}b^{ac}c^{ab}=729$

was one of the conditions rather than

$a^{bc}+b^{ac}+c^{ab}=729$

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    $\begingroup$ Well, yes! You are right and thank you for your attention. There must be a misprint in the book. I tried solving again with the error in the question resolved. If solved with the edits, the correct answer is the fourth option. I have also reported this issue to the publisher of the book. $\endgroup$ May 4 at 8:30

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