Question

How do you simplify `(-27/8)^(-4/3)`?

Answer 1

`(-27/8)^(-4/3) = 16/81`

Explanation:

General rules we need to know:
[1]`color(white)("XXXX")b^(-p) = 1/(b^p)`

[2]`color(white)("XXXX")b^(q/r) = (b^(1/r))^q`

[3]`color(white)("XXXX")b^(1/r) = root(r)(b)`

[4]`color(white)("XXXX")root(r)(b/a) = root(r)(b)/root(r)(a)`

Given the above:
`color(white)("XXXXXXXX")(-27/8)^(-4/3) = (-8/27)^(4/3color(white)("XXXX")`Rule [1]

`color(white)("XXXX")(-8/27)^(4/3) =( (-8/27)^(1/3))^4color(white)("XXXXX")`Rule [2]

`color(white)("XXXX")( (-8/27)^(1/3))^4 = (-root(3)(8)/root(3)(27))^4color(white)("XXXX")`Rules [3] and [4]

`color(white)("XXXX")(-root(3)(8)/root(3)(27))^4 = (-2/3)^4color(white)("XXXXXXX")`Root extraction by observation

`color(white)("XXXX")(-2/3)^4 = 16/81color(white)("XXXXXXXXXXXXX")`Repeated multiplication