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 ${{\left( {{y}^{z}} \right)}^{y}}$

& \therefore {{\left( {{y}^{z}} \right)}^{y}}=z \\\ & {{y}^{z.y}}=z \\\ & \Rightarrow {{y}^{zy}}=z.......(4) \\\ \end{aligned}$$ From equation (3) we know that $$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.