Question:`81^(3/4)` 

`(3^4)^(3/4)` = `3^(4 xx 3/4)` =`3^3` = 27

Question:`(1/64)^(-5/6)` 

`(1/64)^(-5/6)`

=`64^(5/6)`

=`(2^6)^(5/6)`

=`2^(6 xx 5/6)`

=`2^5`

=32

Question:`256^(-1/4` 

`256^(-1/4)`

=`(1/256)^(1/4)`

=`[(1/4)^4]^(1/4)`

=`(1/4)^(4 xx 1/4)`

=`1/4`

Question:`(x^b/x^c)^ ((b^2 + c^2 + bc))` x `(x^c/x^a)^ ((c^2 + a^2 + ca))` x `(x^a/x^b)^ ((a^2 + b^2 + ab))` 

=`(x^(b - c))^((b^2 + c^2 + bc))` x `(x^(c - a))^((c^2 + a^2 + ca))` x `(x^(a - b))^((a^2 + b^2 + ab))`

=`x^((b - c)(b^2 + c^2 + bc))`    x  `x^((c - a)(c^2 + a^2 + ca))`    x  `x^((a - b)(a^2 + b^2 + ab))`

= `x^((b^3 - c^3))`  x `x^((c^3 - a^3))`  x `x^((a^3 - b^3))`

=`x^((b^3 - c^3) + (c^3 - a^3) + (a^3 - b^3))`

=`x^0`

=1

Question:`((2^(1/2) xx 3^(1/3) xx 4^(1/4))/(10^(- 1/5) xx 5^(3/5))// (3^(4/3) xx 5^(- 7/5))/(4^(- 3/5) xx 6))` 

`(2^(1/2) xx 3^(1/3) xx 4^(1/4))/(10^(- 1/5) xx 5^(3/5)) xx (4^(- 3/5) xx 6)/(3^(4/3) xx 5^(- 7/5))`

= `(2^(1/2) xx 3^(1/3) xx 2^(1/2))/(2^(- 1/5) xx 5^(- 1/5) xx 5^(3/5)) xx ((2^2)^(- 3/5) xx (2 xx 3))/(3^(4/3) xx 5^(- 7/5))`

= `(2^(1/2) xx 3^(1/3) xx 2^(1/2))/(2^(- 1/5) xx 5^(- 1/5) xx 5^(3/5)) xx (2^(- 6/5) xx 2 xx 3)/(3^(4/3) xx 5^(- 7/5))`

= `(2^(1/2 + 1/2 +1/5 - 6/5 +1) . 3^(1/3 +1 - 4/3))/5^(-1/5 + 3/5 - 7/5)`

= `(2^1 xx 3^0 xx 5^1)`

= (2 x 1 x 5)
= 10

Question:Which is largest: `root(3)(4)` or `root(4)(5)` 

`:. \ root(3)(4)`

 = `4^(1/3)`

 = `4^((1/3 xx 3/3))`

 = `4^(4/12)`

 = `(4^4)^(1/12)`

 = `256^(1/12)`


`:. \ root(4)(5)`

 = `5^(1/4)`

 = `4^((1/4 xx 3/3))`

 = `5^(3/12)`

 = `(4^3)^(1/12)`

 = `125^(1/12)`

 Clearly. `(256)^(1/12)` > `(125)^(1/12)`

 or `root(3)(4)` > `root(4)(5)`