Question

How do you simplify `((x^-3)^4x^4)/(2x^-3)` and write it using only positive exponents?

Answer 1

See the entire simplification process below:

Explanation:

First, use the rule of exponents to simplify the term in parenthesis in the numerator: `(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))`

`((x^color(red)(-3))^color(blue)(4)x^4)/(2x^-3) -> (x^(color(red)(-3) xx color(blue)(4))x^4)/(2x^-3) -> (x^-12x^4)/(2x^-3)`

Next, use this rule for exponents to simplify the numerator:

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

`(x^color(red)(-12) x^color(blue)(4))/(2x^-3) -> x^(color(red)(-12) +color(blue)(4))/(2x^-3) -> x^-8/(2x^-3)`

Now, use this rule of exponents to complete the simplification using only positive exponents: `x^color(red)(a)/x^color(blue)(b) = 1/x^(color(blue)(b)-color(red)(a))`

`x^color(red)(-8)/(2x^color(blue)(-3)) -> 1/(2x^(color(blue)(-3)-color(red)(-8))) -> 1/(2x^(color(blue)(-3)+color(red)(8))) -> 1/(2x^5)`

Answer 2

`1/(2x^5)`

Explanation:

`((x^-3)^4 x^4)/(2x^-3) = (x^-12*x^4)/(2x^-3) = x^-8/(2x^-3) =(x^-5*cancel(x^-3))/ (2*cancel(x^-3))=x^-5/2=1/(2x^5)` [Ans]