simplify
`(i) (1^(3)+2^(3)+3^(3))^(1/2)`(ii)`(3/5)^(4)(8/5)^(-12)(32/5)^(6)`,(iii)`(1/27)^(-2/3)`
(vi)`[((625)^(-1/2)))^(-1/4)]^(2)`,(v)`(9^(1/3)xx27^(-1/2))/(3^(1/6)xx3^(-2/3))`(vi)`64^(-1/3)[64^(1/3)-64^(2/3)]`
`(8^(1/3)xx16^(1/3))/(32^(-1/3)`

Explanation

`(i)(1^(3)+2^(3)+3^(3))^(1/2)=(1+8+27)^(1/2)`
`(36)^(1/2)=(6^(2))^(1/2)=6^(2xx1/2)=6`
(ii) `(3/4)^(4)(8/5)^(-12)(32/5)^(6)=(3^(4))/(5^(4))xx(5/2^(3))^(12)xx(2^(5)/5)^(6)(a^(-1)=1/a)`
`(3^(4))/(5^(4))xx5^(12)/2^(36)xx2^(30)/5^(6) " " [(a^(m))^(n)=a^(mn)]`
`(3^(4)xx5^(12-4-6))/(2^(36-30)) " " [a^(m)/a^(n)=a^(m-n)]`
`(3^(4))/2^(6)xx5^(2)=(81xx25)/64=(2025)/64`
(iii) `(1/27)^(-2//3)=(1/3^(3))^(-2//3)=(3^(-3))^(-2//3)`
`3^(-3xx2/3)=3^(2)=9 [(a^(m)^(n)=a^(mn)]`
(iv) `[((625)^(-1/4))^(-1/4)]^(2)=[((25^(-1))^(-1/2))^(-1/4)]^(2)=(25^(-1))^(1/4xx2)=[(5^(2))^(-1)]^(1/4xx2) " " [(a^(m))^(n)=a^(mn)]`
`5^(-2xx-1/4xx2)=5^(1)=5)`
(v) `(9^(1/3)xx27^(-1/2))/(3^(1/6)xx3^(-2/3))=((3^(2))^(1/3)xx(3^(3))^(1/2))/(3^(1/6)xx3^(-2/3))`
`(3^(2/3)xx3^(-3/2))/(3.^(1/6)xx3^(-2/3))=(3^(2/3-3/2))/(3^(1/6-2/3))`
`(3^((4-9)/6))/(3^((1-4)/6))=(3^(-5/6))/(3^(-3/6))=3^(-5/6+3/6)=3^(-2/6)=3^(1/3 " " [a^(m)/a^(n)=a^(m-n)]`
(vi) `64^(-1/3)[64^(1/3)-64^(2/3)]=[(4^(3))^(1/3)-(4^(3))^(2/3)]`
`=4^(3xx-1/3)(4^(3xx1/3)-4^(3xx2/3))=4^(-1)(4-4^(2)) " " [(a^(m))^(n)=a^(mn)]`
`1/4(4-16)=-12/4=-3`
(vii) `(8^(1/3)xx16^(1/3))/(32^(-1/3))=((2^(3))^(1/3)xx(2^(4))^(1/3))/((2^(5))^(-1/3))=(2^3xx1/3xx2^(4xx1/3))/(2^(5xx-1/3))`
`(2^(3/3+4/3))/(2^(-5/3))=2^(7/3)/2^(-5/3)=2^(7/3+5/3)=2^(12/3)=2^(4)=16 " " [(a^(m))/a^(n)=a^(m-n)]`