Question

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

Answer 1

See full simplification process below

Explanation:

First, we we use this rule for exponents to start the simplification:

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

`(4x^4y^-3)^-2 ->`

`4^-2x^(4 xx -2)y^(-3 xx -2) ->`

`4^-2x^-8y^6`

Now, we will use this rule for exponents to finalize the simplification leaving only terms with positive exponents:

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

`y^6/(4^(- -2)x^(- -8)`

`y^6/(4^2x^8`

`y^6/(16x^8)`

Answer 2

`(y^6)/(16x^8)`

Explanation:

Using the `color(blue)"laws of exponents"`

`color(red)(bar(ul(|color(white)(2/2)color(black)((a^mb^n)^p=a^(mp)b^(np))color(white)(2/2)|)))`
This can be extended to more than 2 products.

`rArr(4^1x^4y^(-3))^(-2)=4^((1xx-2))x^((4xx-2))y^((-3xx-2))`

`=4^(-2)x^(-8)y^6`

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(a^-mhArr1/a^m)color(white)(2/2)|)))`

`rArr4^(-2)x^(-8)y^6=1/4^2xx1/x^8xxy^6/1`

`=(1xx1xxy^6)/(16xxx^8xx1)=(y^6)/(16x^8)`