1.

without actually calculating , the cubes , find the valuse of (i) `((1)/(2))^(3)+((1)/(3))^(3)-((5)/(6))^(3)`(ii) `(0.2)^(3)-(0.3)^(3)+(0.1)^(3)`

Answer» given , `((1)/(2))^(3)+((1)/(3))^(3)-((5)/(6))^(3)or ((1)/(2))^(3)+((1)/(3))^(3)+(-(5)/(6))^(3)`
here , we see that , `(1)/(2) +(1)/(3)-(5)/(6)=(3+2-5)/(6)=(5-5)/(6)=0`
`therefore ((1)/(2))^(3)+((1)/(3))^(3)-((5)/(6))^(3)=3xx(1)/(2)xx(1)/(3)xx(-(5)/(6))=-(5)/(12)`
[using identity , if a+b+c=0 then `a^(3)+b^(3)+c^(3)`= 3abc]
(ii) given `(0.2)^(3)-(0.3)^(3)+(0.1)^(3)or (0.2)^(3)+(-0.3)^(3)+(0.1^(3)`
here , we see that , `0.2-0.3+0(.1)^(3)=3xx(0.2)xx(0.3)xx(0.1)`
[using identity if `a+b+c =0` then `a^(3)+b^(3)+c^(3)=3abc`]
`=-0.6xx0.03=-0.018`


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