If $a^{1/a}=b^{1/b}=c^{1/c}$ and $a^{bc}+b^{ac}+c^{ab}=729$, find the value of $a^{1/a}$

Question

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. $\sqrt[abc]{9}$

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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!

Answer 1

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

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