`(1000)^7 -:  10^18` =  `(1000)^7/10^18`   =  `(10^3)^7/10^18` =  `10^((3 xx 7))/(10^18)`

=`10^21/10^18` = `(10)^(21 - 18)` =  `10^3` =  1000

`(1/216)^(- (2)/(3))  -:  (1/27)^(- (4)/(3)) `   =  `(216)^(2/3)  -:  (27)^(4/3)`      = `((216)^(2/3))/((27)^(4/3))`

= `((6^3)^(2/3))/((3^3)^(4/3)`  = `(6^((3 xx 2/3)))/(3^((3 xx 4/3)))`   =  `(6^2)/(3^4)`=  `36/81` =  `4/9`

`(2.4 xx 10^3) -: (8 xx 10^-2)`= `(2.4 xx 10^3)/(8 xx 10^-2)` = `(24 xx 10^2)/(8 xx 10^-2)`   = `(3 xx 10^4)`

` (10)^150 -: (10)^146`   =  `((10)^150)/((10)^146)`     =   `(10)^(150 - 146)`     = `10^4` =  10000

`(1)/((216)^(-  2/3)) + (1)/((256)^(-  3/4))  +(1)/((32)^(-  1/5)) `

  =`(1)/( (6^3)^( - 2/3))  + (1)/( (4^4)^( - 3/4))  + (1)/( (2^5)^( - 1/5)`

= `(1)/6^(3  xx  ( - 2)/3) +(1)/4^(4  xx (  - 3)/4) +(1)/2^(5  xx  (- 1)/5) `

=`(1)/(6)^-2 + (1)/(4)^-3 + (1)/(2)^-1`=  `(6^2 + 4^3 +2^1)` =  (36 +64 + 2) = 102.

`5^(1/4)  xx (125)^0.25`  =` 5^0.25 xx (5^3)^0.25` =  ` 5^0.25 xx 5^(3 xx 0.25)` = `5^0.25 xx 5^0.75`
= `5^((0.25 + 0.75))` = `5^1` = 5.

`( - 1/216)^(  - (2)/(3)) `=` [( - 1/6)^3] ^(- (2)/(3)`         = `(- 1/6)^(3 xx((-2))/(3))`     =  `(- 1/6)^ - 2`

= `1/(- 1/6)^2`       =  `1/(1/36)`     = 36.

`(32/243)^(-  (4)/(5))` =  `{(2/3)^5}^( - (4)/(5))`  = `((2)/(3))^(5 xx ((- 4))/(5))`  = `((2)/(3))^(- 4)`

= `(3/2)^4`    =`(3^4)/(2^4)`=    `81/16`

`(sqrt(8))^((1)/(3))` = `(8^(1/2))^((1)/(3))`        =  `8^((1/2 xx 1/3))`       = ` 8 ^((1)/(    6))`   = `(2^3)^((1)/(6))`

  =     `2^((3 xx 1/6))`   = `2^(1/2)`        = `sqrt(2)`

`(256)^((5)/(4))` =  `(4^4)^((5)/(4))` =   `4^((4 xx 5/4))` =  `4^5` =  1024.