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Question
Show that:
`(a^(x+1)/a^(y+1))^(x+y)(a^(y+2)/a^(z+2))^(y+z)(a^(z+3)/a^(x+3))^(z+x)=1`
Solution
`(a^(x+1)/a^(y+1))^(x+y)(a^(y+2)/a^(z+2))^(y+z)(a^(z+3)/a^(x+3))^(z+x)=1`
LHS = `(a^(x+1)/a^(y+1))^(x+y)(a^(y+2)/a^(z+2))^(y+z)(a^(z+3)/a^(x+3))^(z+x)`
`=(a^(x+1-y-1))^(x+y)(a^(y+2-z-2))^(y+z)(a^(z+3-x-3))^(z+x)`
`=(a^(x-y))^(x+y)(a^(y-z))^(y+z)(a^(z-x))^(z+x)`
`=(a^((x-y)(x+y)))(a^((y-z)(y+z)))(a^((z-x)(z+x)))`
`=(a^(x^2-y^2))(a^(y^2-z^2))(a^(z^2-x^2))`
`=a^(x^2-y^2+y^2-z^2+z^2-x^2)`
`=a^0`
= 1
= RHS
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