Question

How do you simplify `(20qr^-2t^-5)/(rq^0r^4t^-2)`?

Answer 1

See a solution process below:

Explanation:

First, rewrite this expression as:

`20(q/q^0)(r^-2/(r * r^4))(t^-5/t^-2)`

Next, simplify the `q` term using this rule of exponents:

`a^color(red)(0) = 1`

`20(q/q^color(red)(0))(r^-2/(r * r^4))(t^-5/t^-2) => 20(q/color(red)(1))(r^-2/(r * r^4))(t^-5/t^-2) =>`

`20q(r^-2/(r * r^4))(t^-5/t^-2)`

Next, use these rules of exponents to simplify the `r` terms:

`a = a^color(red)(1)` and `x^color(red)(a) xx 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))`

`20q(r^-2/(r * r^4))(t^-5/t^-2) => 20q(r^-2/(r^color(red)(1) * r^color(blue)(4)))(t^-5/t^-2) =>`

`20q(r^-2/(r^(color(red)(1) + color(blue)(4))))(t^-5/t^-2) => 20q(r^color(red)(-2)/(r^color(blue)(5)))(t^-5/t^-2) =>`

`20q(1/(r^(color(blue)(5) - color(red)(-2))))(t^-5/t^-2) => 20q(1/(r^(color(blue)(5) + color(red)(2))))(t^-5/t^-2) =>`

`20q(1/r^7)(t^-5/t^-2) => (20q)/r^7(t^-5/t^-2)`

Now, use this rule to simplify the `t` terms:

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

`(20q)/r^7(t^color(red)(-5)/t^color(blue)(-2)) => (20q)/r^7(1/t^(color(blue)(-2)-color(red)(-5))) => (20q)/r^7(1/t^(color(blue)(-2)+color(red)(5))) =>`

`(20q)/r^7(1/t^3) => (20q)/(r^7t^3)`