Question

If \[{{x}^{y}}=z,{{y}^{z}}=x,{{z}^{x}}=y\], then find xyz.
A. 2
B. 1
C. 4
D. 3

Answer 1

Hint:Take the first expression, \[{{x}^{y}}=z\], substitute the value of x and obtain a new expression, in this substitute value of y and solve it using the power rule of exponents.

Complete step-by-step answer:

We have been given three expressions,
\[{{x}^{y}}=z......(1)\]
\[{{y}^{z}}=x.....(2)\]
\[{{z}^{x}}=y.....(3)\]
Now let us put the value of \[{{x}^{y}}=z\] in equation (1).
\[{{x}^{y}}=z\], put \[x={{y}^{z}}\].
\[{{\left( {{y}^{z}} \right)}^{y}}=z\]
By power rule of exponents we know that to raise a power to power, just multiply the exponents, which is of the form,
\[{{\left( {{x}^{m}} \right)}^{n}}={{x}^{mn}}\].
Similarly, we can apply power rule in \[{{\left( {{y}^{z}} \right)}^{y}}\]
\[\begin{align}
  & \therefore {{\left( {{y}^{z}} \right)}^{y}}=z \\
 & {{y}^{z.y}}=z \\
 & \Rightarrow {{y}^{zy}}=z.......(4) \\
\end{align}\]
From equation (3) we know that \[y={{z}^{x}}\].
Let us substitute this value of y in equation (4).
\[{{\left( {{z}^{x}} \right)}^{zy}}=z\].
Let us apply exponent power rule again on \[{{\left( {{z}^{x}} \right)}^{xy}}\], we get,
\[{{z}^{xyz}}=z\].
Now this is of the form \[{{z}^{xyz}}={{z}^{1}}\].
Thus according to the law of indices, \[{{A}^{B}}={{A}^{C}}\Rightarrow B=C.\]
Similarly, \[{{z}^{xyz}}={{z}^{1}}\]
\[\Rightarrow xyz=1\].
Thus we got the value of xyz = 1.
Option B is the correct answer.

Note: In a question like this remember that substitution works here. We assumed that \[{{x}^{y}}=z\] and then substituted the values of x and y to it. You can also take the term \[{{y}^{z}}=x\] and substitute the value of y and z to it or using \[{{z}^{x}}=y\]. Either way we have to do substitution to get the required number.