Question

How do you simplify `( 9 / 49) ^ (- 3 / 2)`?

Answer 1

`=27/(343`

Explanation:

As per property:
`(a/b)^color(blue)(m)= a^color(blue)(m)/(b^color(blue)(m`

Applying the above to the expression :

`(9/49)^ (-3/2) = 9^color(blue)(-3/2)/(49^color(blue)(-3/2`

`(3^2)^(color(blue)(-3/2))/((7^2)^color(blue)(-3/2`

`=(3^cancel2)^(-3/cancel2)/((7^cancel2)^(-3/cancel2)`

`color(blue)("~~~~~~~~~~~~~~Tony B Formatting test~~~~~~~~~~~~~~~~~~~")`
`(3^(cancel(2))) (3/(cancel(2)))`

`(3^(cancel(2)))^(3/(cancel(2)))`

`color(red)("The formatting code can not cope with changing the second")` `color(red)("bracket group into index form.")`
`color(blue)("'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")`

`=3^-3/(7^-3`

`=(1/27)/(1/343)`

`=343/27`

Answer 2

`(9/49)^(-3/2)=[(3/7)^2]^(-3/2)=(3/7)^-3=(7/3)^3=343/27`

Explanation:

The minus in front of the index is instruction that this is a reciprocal

So we have: `1/((9/49)^(3/2))`

This is `((49)^(3/2))/((9)^(3/2))`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider `color(white)(..)9^(3/2)`

This is the same as `(sqrt(9)color(white)(.) )^3=3^3=27`

Giving: `((49)^(3/2))/27`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider: `49^(3/2)`

This is the same as `(sqrt(49))^3=7^3=343`

Giving:`(343)/27 = 12 19/27`