यदि परिमेय संख्याएँ तथा एक धनात्मक वास्तविक संख्या है तब सिद्ध कीजिए कि-
`(i) ((x^(a))/(x^(b)))^(a^(2)+ab+b^(2))*((x^(b))/(x^(c )))^(b^(2)+bc+c^(2))*((x^(c ))/(x^(a)))^(c^(2)+ca+a^(2))=1`
`(ii) ((x^(a))/(x^(b)))^(a+b)*((x^(b))/(x^(c )))^(b+c)*((x^(c ))/(x^(a)))^(c+a)=1`

Explanation

`(i)` बाया पक्ष `=((x^(a))/(x^(b)))^(a^(2)+ab+b^(2))*((x^(b))/(x^(c )))^(b^(2)+bc+c^(2))*((x^(c ))/(x^(a)))^(c^(2)+ca+a^(2))`
`=(x^(a-b))^(a^(2)+ab+b^(2))*(x^(b-c))^(b^(2)+bc+c^(2))*(x^(c-a))^(c^(2)+ca+a^(2))`
`=(x)^((a-b)(a^(2)+ab+b^(2)))*(x)^((b-c)(b^(2)+bc+c^(2)))*(x)^((c-a)(c^(2)+ca+a^(2)))`
`=x^(a^(3)-b^(3))*x^(b^(3)-c^(3))*x^(c^(3)-a^(3))=x^(a^(3)-b^(3)+b^(3)-c^(3)+c^(3)-a^(3))=x^(0)=1=` दाया पक्ष
`(ii)` बाया पक्ष `=((x^(a))/(x^(b)))^(a+b)*((x^(b))/(x^(c )))^(b+c)*((x^(c ))/(x^(a)))^(c+a)`
`=(x^(a-b))^(a+b)**(x^(b-c))^(b+c)(x^(c-a))^(c+a)`
`=x^((a-b)(a+b))*x^((b-c)(b+c))*x^((c-a)(c+a))`
`=x^(a^(2)-b^(2))*x^(b^(2)-c^(2))*x^(c^(2)-a^(2))=x^(a^(2)-b^(2)+b^(2)-c^(2)+c^(2)-a^(2))=x^(0)=1` दाया पक्ष