यदि `x` कोई धनात्मक वास्तविक संख्या तथा `a,b,c` परिमेय संख्याये है तो सिद्ध कीजिए कि
`(i) ((x^(b))/(x^(c )))^(a)((x^(c ))/(x^(a)))^(b)((x^(a))/(x^(b)))^(c )=1` `(ii) ((x^(a))/(x^(b)))^(1//ab)((x^(b))/(x^(c )))^(1//bc)((x^(c ))/(x^(a)))^(1//ac )=1`

Explanation

`(i)` यहाँ
बाया पक्ष `=((x^(b))/(x^(c )))^(a)*((x^(c ))/(x^(a)))^(b)*((x^(a))/(x^(b)))^(c )`
`=(x^(b-c))^(a)*(x^(c-a))^(b)*(x^(a-b))^(c)=x^(ab-ac)*x^(bc-ba)*x^(ac-bc)`
`=x^(ab-ac+bc-ba+ac-bc)=x^(0)=1=` दाया पक्ष
`(ii)` बाया पक्ष `=((x^(a))/(x^(b)))^(1//ab)((x^(b))/(x^(c )))^(1//bc)((x^(c ))/(x^(a)))^(1//ac )`
`=(x^(a-b))^(1//ab)*(x^(b-c))^(1//bc)*(x^(c-a))^(1//ac)`
`=x^((a-b)//ab)*x^((b-c)//bc)*x^((c-a)//ac)`
`=x^((1)/(b)-(1)/(a))*x^((1)/(c)-(1)/(b))*x^((1)/(a)-(1)/(c ))=x^((1)/(b)-(1)/(a)+(1)/(c )-(1)/(b)+(1)/(a)-(1)/(c ))=x^(0)=1` दाया पक्ष