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