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# Prove that (x^{a }/ x^{b})^{1/ab} (x^{b} / x^{c})^{1/bc} (x^{c }/ x^{a})^{1/ca} = 1

Exponential formulas of multiplication and division can be used to simplify such expressions.

## Answer: Value of the L.H.S. comes out to be 1 thus the identity holds true.

Go through the proof and understand the solvation of this expression.

**Explanation:**

L.H.S = (x^{a }/ x^{b})^{1/ab} (x^{b} / x^{c})^{1/bc} (x^{c }/ x^{a})^{1/ca}

On simplifying the L.H.S, we get:

(x^{a }/ x^{b})^{1/ab} (x^{b} / x^{c})^{1/bc} (x^{c }/ x^{a})^{1/ca}

⇒ (x^{a - b})^{1/ab }(x^{b - c})^{1/bc }(x^{c - a})^{1/ca }

⇒ (x^{(a - b)/ab})^{ }(x^{(b - c)/bc})^{ }(x^{(c - a)/ca})

⇒ (x^{[(a - b)/ab] + [(b - c)/bc] + [(c - a)/ca]})

⇒ x ^{(ac - bc + ab - ca + bc - ab) / abc}

⇒ x^{0 }= 1

= R.H.S

### Since, R.H.S = L.H.S, hence proved.

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