Question
For any positive real number x, write the value of $\big\{(\text{x}^\text{a})^\text{b}\big\}^{\frac{1}{\text{ab}}}\big\{(\text{x}^\text{b})^\text{c}\big\}^{\frac{1}{\text{bc}}}\big\{(\text{x}^\text{c})\text{a}\big\}^{\frac{1}{\text{ca}}}$
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Answer
We have to simplify $\big\{(\text{x}^\text{a})^\text{b}\big\}^{\frac{1}{\text{ab}}}\big\{(\text{x}^\text{b})^\text{c}\big\}^{\frac{1}{\text{bc}}}\big\{(\text{x}^\text{c})\text{a}\big\}^{\frac{1}{\text{ca}}}$
So, $\big\{(\text{x}^\text{a})^\text{b}\big\}^{\frac{1}{\text{ab}}}\big\{(\text{x}^\text{b})^\text{c}\big\}^{\frac{1}{\text{bc}}}\big\{(\text{x}^\text{c})\text{a}\big\}^{\frac{1}{\text{ca}}}$
$=\big\{\text{x}^{\text{ab}}\big\}^{\frac{1}{\text{ab}}}\big\{\text{x}^{\text{bc}}\big\}^{\frac{1}{\text{bc}}}\big\{\text{x}^{\text{ca}}\big\}^{\frac{1}{\text{ca}}}$
$=\text{x}^{\text{ab}\times\frac{1}{\text{ab}}}\times\text{x}^{\text{bc}\times\frac{1}{\text{bc}}}\times\text{x}^{\text{ca}\times\frac{1}{\text{ca}}}$
$=\text{x}^1\times\text{x}^1\times\text{x}^1$ By using rational exponents $\text{a}^\text{m}\times\text{a}^\text{n}=\text{a}^{\text{m}+\text{n}},$
we get $\Big\{\big(\text{x}^{\text{a}}\big)^\text{b}\Big\}^{\frac{1}{\text{ab}}}\Big\{\big(\text{x}^\text{b}\big)^\text{c}\Big\}^{\frac{1}{\text{bc}}}\Big\{\big(\text{x}^\text{c}\big)^\text{a}\Big\}^{\frac{1}{\text{ca}}}=\text{x}^{1+1+1}$
$=\text{x}^3$
Hence the value of $\big\{(\text{x}^\text{a})^\text{b}\big\}^{\frac{1}{\text{ab}}}\big\{(\text{x}^\text{b})^\text{c}\big\}^{\frac{1}{\text{bc}}}\big\{(\text{x}^\text{c})\text{a}\big\}^{\frac{1}{\text{ca}}}$ is $\text{x}^3$
So, $\big\{(\text{x}^\text{a})^\text{b}\big\}^{\frac{1}{\text{ab}}}\big\{(\text{x}^\text{b})^\text{c}\big\}^{\frac{1}{\text{bc}}}\big\{(\text{x}^\text{c})\text{a}\big\}^{\frac{1}{\text{ca}}}$
$=\big\{\text{x}^{\text{ab}}\big\}^{\frac{1}{\text{ab}}}\big\{\text{x}^{\text{bc}}\big\}^{\frac{1}{\text{bc}}}\big\{\text{x}^{\text{ca}}\big\}^{\frac{1}{\text{ca}}}$
$=\text{x}^{\text{ab}\times\frac{1}{\text{ab}}}\times\text{x}^{\text{bc}\times\frac{1}{\text{bc}}}\times\text{x}^{\text{ca}\times\frac{1}{\text{ca}}}$
$=\text{x}^1\times\text{x}^1\times\text{x}^1$ By using rational exponents $\text{a}^\text{m}\times\text{a}^\text{n}=\text{a}^{\text{m}+\text{n}},$
we get $\Big\{\big(\text{x}^{\text{a}}\big)^\text{b}\Big\}^{\frac{1}{\text{ab}}}\Big\{\big(\text{x}^\text{b}\big)^\text{c}\Big\}^{\frac{1}{\text{bc}}}\Big\{\big(\text{x}^\text{c}\big)^\text{a}\Big\}^{\frac{1}{\text{ca}}}=\text{x}^{1+1+1}$
$=\text{x}^3$
Hence the value of $\big\{(\text{x}^\text{a})^\text{b}\big\}^{\frac{1}{\text{ab}}}\big\{(\text{x}^\text{b})^\text{c}\big\}^{\frac{1}{\text{bc}}}\big\{(\text{x}^\text{c})\text{a}\big\}^{\frac{1}{\text{ca}}}$ is $\text{x}^3$
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