Question

How do you simplify `((2r^3)/s)^2(s/r^3)^3` and write it using only positive exponents?

Answer 1

`((2r^3)/s)^2(s/r^3)^3=(4s)/(r^3)`

Explanation:

`((2r^3)/s)^2(s/r^3)^3 = (2^2r^(3*2))/(s^2)*s^3/r^(3*3)`

`=(4r^6)/s^2*s^3/r^9`

`= (4s^(3-2))/(r^(9-6))`

`=(4s)/(r^3)`

Answer 2

See the entire simplification process below:

Explanation:

First use these two rules for exponents to begin simplifying:

`a = a^color(red)(1)`

`(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))`

`((2r^3)/s)^2(s/r^3)^3 -> ((2^color(red)(1)r^color(red)(3))/s^color(red)(1))^color(blue)(2)(s^color(red)(1)/r^color(red)(3))^color(blue)(3) ->`

`(2^(color(red)(1) xx color(blue)(2))r^(color(red)(3) xx color(blue)(2)))/s^(color(red)(1) xx color(blue)(2)) xx s^(color(red)(1) xx color(blue)(3)) /r^(color(red)(3) xx color(blue)(3)) -> (2^2r^6s^3)/(s^2r^9) -> (4r^6s^3)/(s^2r^9)`

Now, use these two rules for exponents to complete the simplification:

`x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))`

`x^color(red)(a)/x^color(blue)(b) = 1/x^(color(blue)(b)-color(red)(a))`

`(4r^color(red)(6)s^color(red)(3))/(s^color(blue)(2)r^color(blue)(9)) -> (4s^(color(red)(3)-color(blue)(2)))/r^(color(blue)(9)-color(red)(6)) -> (4s^1)/r^3 -> (4s)/r^3`