1.

(a) Find the RF of `2^(1/3)+ 2^(-1/3)`(b) Find the RF of `5^(1/3)-5^(-1/3)`

Answer» (a) `2^(1//3)+2^(-1//3)`
Let a`=2^(1//3) and b =2^(-1//3)`
`a^(3)=(2^(1//3))^(3)=2`
`b^(3)=(2^(-1//3))^(3)=2^(-1)=(1)/(2)`
But `a^(3)+b^(3)=(a+b)(a^(2)-ab+b^(2))`
`therefore a^(2)-ab+b^(2)=(2^(1//3))^(2)-(2^(1//3).2^(-1//3))+(2^(-1//3)+(2^(-1//3))^(2)=2^(2//3)-1+2^(-2//3)`
`therefore ` RF of `2^(1//3)+2^(-1//3)` is `2^(2//3)-1+2^(-2//3)`
(b) `5^(1//3)-5^(-1//3)`
We have `a^(3)-b^(3)=(a-b)(a^(2)+ab+b^(2))`
`therefore ` RF of `5^(1//3)-5^(-1//3)` is `[5^(1//3)]^(2)+[5^(1//3).5^(-1//3)]+[5^(-1//3)]^(2)`
`=5^(2//3)+1+5^(-2//3)`


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