Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 2.3 Junior Find the answer

If $a^{x}= c^{q}= b$ and $c^{y}= a^{z}= d$ , then
$\textbf{(A)}\ xy = qz$ $\textbf{(B)}\ \frac{x}{y}=\frac{q}{z}$ $\textbf{(C)}\ x+y = q+z$ $\textbf{(D)}\ x-y = q-z$
$\textbf{(E)}\ x^{y}= q^{z}$

Your answer

Multiple choice: answer with the letter of the option you want.

Solution

Try solving both equations for $a$ . Taking the $x$ -th root of both sides in the first equation and the $z$ -the root of both sides in the second gives $a=c^{\frac{q}x}$ and $a=c^{\frac{y}z}$ .
So $\frac{q}x=\frac{y}z$ . Multiplying both sides by $xz$ , $qz=xy$ . $\boxed{\textbf{(A)}}$ .

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