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


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Answer:

^@\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 }^@.

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