### If $a^x = \sqrt{b}$, $b^y = \sqrt{c}$ and $c^z = \sqrt{a}$, find the value of $xyz$.

$\dfrac{1}{ 8 }$

Step by Step Explanation:
1. Let's write $a^x = \sqrt{b}$ as:
\begin{align} & a^x = b^{ 1\over 2 } \\ \implies & a^{ 2x } = b \end{align}
and
\begin{align} & b^y = \sqrt{c} \\ \implies & b^y = c^{ 1 \over 2 } \\ \implies & b^{ 2y } = c \end{align}
and
\begin{align} & c^z = \sqrt{a} \\ \implies & c^z = a^{ 1 \over 2 } \\ \implies & c^{ 2z } = a \end{align}
2. Put the value of $c$ in $c^{ 2z } = a$
\begin{align} &\implies a = \left(b^{ 2y } \right)^{ 2z } \\ &\implies a = b^{ 2 \times 2 \times yz } \\ &\implies a = b^{ 4yz } \end{align}
Now put the value of $b$
\begin{align} & a = \left(a^{ 2x } \right)^{ 4yz } \\ \implies & a = a^{ 2 \times 4 \times xyz } \\ \implies & a^1 = a^{ 8xyz } \end{align}
3. On comparing powers in above equation,
$\implies 1 = 8xyz \\ \implies xyz = \dfrac{1}{ 8 }$
4. Therefore, the value of xyz is $\dfrac{1}{ 8 }$.