If ^@a^x = \sqrt{b} ^@, ^@b^y = \sqrt{c} ^@ and ^@c^z = \sqrt{a} | Math Question and Answer | Edugain Singapore

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

Answer:

$\dfrac{1}{ 8 }$

Step by Step Explanation:

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
$\begin{align} & c^z = \sqrt{a} \\ \implies & c^z = a^{ 1 \over 2 } \\ \implies & c^{ 2z } = a \end{align}$
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 $b$
$\begin{align} & a = \left(a^{ 2x } \right)^{ 4yz } \\ \implies & a = a^{ 2 \times 4 \times xyz } \\ \implies & a^1 = a^{ 8xyz } \end{align}$
On comparing powers in above equation,
$\implies 1 = 8xyz \\ \implies xyz = \dfrac{1}{ 8 }$
Therefore, the value of xyz is $\dfrac{1}{ 8 }$ .

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