Question

For any positive real number x, write the value of  \[\left\{ \left( x^a \right)^b \right\}^\frac{1}{ab} \left\{ \left( x^b \right)^c \right\}^\frac{1}{bc} \left\{ \left( x^c \right)^a \right\}^\frac{1}{ca}\]

Answer 1

\[\left\{ \left( x^a \right)^b \right\}^\frac{1}{ab} \left\{ \left( x^b \right)^c \right\}^\frac{1}{bc} \left\{ \left( x^c \right)^a \right\}^\frac{1}{ca}\] So,

\[\left\{ \left( x^a \right)^b \right\}^\frac{1}{ab} \left\{ \left( x^b \right)^c \right\}^\frac{1}{bc} \left\{ \left( x^c \right)^a \right\}^\frac{1}{ca}\] = `{x^(ab)}^(1/(ab)) {x^(bc)}^(1/(bc)) {x^(ca)}^(1/(ca))`

=`x^(ab xx 1/(ab)) xx x^(bc xx 1/(bc) xx x^(ac xx 1/(ca)))`

= `x^(ab xx 1/(ab)) xx x^(bc xx 1/(bc) xx x^(ac xx 1/(ca)))`

`= x^1 xx x^1 xx x^1`

By using rational exponents `a^m xx a^n xx a^(m+n), ` we get

\[\left\{ \left( x^a \right)^b \right\}^\frac{1}{ab} \left\{ \left( x^b \right)^c \right\}^\frac{1}{bc} \left\{ \left( x^c \right)^a \right\}^\frac{1}{ca}\] = `x^(1+1+1)`

`=x^3`

Hence the value of

\[\left\{ \left( x^a \right)^b \right\}^\frac{1}{ab} \left\{ \left( x^b \right)^c \right\}^\frac{1}{bc} \left\{ \left( x^c \right)^a \right\}^\frac{1}{ca}\] = `x^(1+1+1)` is `=x^3`