Question

How do you simplify `81^(-3/2)`?

Answer 1

`81^(-3/2)=1/729`

Explanation:

`81^(-3/2)`

= `(3^4)^(-3/2)`

= `3^((4xx-3/2))`

= `3^(2xx-3)`

= `3^(-6)`

= `1/3^6`

= `1/729`

Answer 2

`1/729`

Explanation:

Recall: 2 laws of indices `" " rarr " "x^-m = 1/x^m`

`color(white)(xxxxxxxxxxxxxxxxx) rarr" " x^(p/q) = rootq (x^p) = (rootq x)^p`

`81^(-3/2) = 1/(sqrt81^3)" "larr` the cube of the square root of 81

=`1/(9^3)`

=`1/729`

Note that 81 is both a square number and a 4th power.
However, the square root can be obtained immediately.

In work on indices, usually only the principal (+) square root is considered.