How can I calculate manually `(9/16)^(-1/2)`?

You will need to use the following formulae:

`(a/b)^(-x)=(b/a)^x`                                                                          (1)

`(a/b)^x=a^x/b^x`                                                                                  (2)

`a^(1/x)=root[x](a)`                                                                                    (3)

Let us now apply those formulae on the problem at hand. First we get rid of the minus sign in the exponent by using the formula (1).

`(9/16)^(-1/2)=(16/9)^(1/2)`

Now we apply formula (2).

`16^(1/2)/9^(1/2)`

Now we apply formula (3) which basically tells us that the power with exponent `1/2`  is the same as taking square root.

`sqrt(16)/sqrt(9)=4/3`

Therefore, the solution is `4/3.`