Question

How do you simplify `((r^2t^-3)/(r^-3t^5))^-8` and write it using only positive exponents?

Answer 1

See below.

Explanation:

Let's simplify inside the expression first.

`\frac(r^2t^-3)(r^-3t^5)=r^5/t^8`

This is to the `-8`th power, so we flip the fraction, and then multiply the exponents by `8`.

=`t^64/r^40`

Answer 2

See the entire solution process below:

Explanation:

First, we will use these two rules of exponents to simplify the terms within the parenthesis:

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

`((r^color(red)(2)t^color(red)(-3))/(r^color(blue)(-3)t^color(blue)(5)))^-8 = (t^(color(red)(-3)-color(blue)(5))/r^(color(blue)(-3)-color(red)(2)))^-8 = (t^-8/r^-5)^-8`

Now, we will use this rule of exponents to eliminate the outer exponent and eliminate the negative exponents:

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

`(t^color(red)(-8)/r^color(red)(-5))^color(blue)(-8) = t^(color(red)(-8) xx color(blue)(-8))/r^(color(red)(-5) xx color(blue)(-8)) = t^64/r^40`