Question

How do you simplify `(\frac { - 2x ^ { 3} y ^ { 0} z } { 4x z ^ { 2} } ) ^ { 5}`?

Answer 1

`(-x^10)/(32z^5)`

Explanation:

For the first simplification we can use the rule for exponents:

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

`((-2x^3color(red)(y^0)z)/(4xz^2))^5 -> ((-2x^3color(red)(1)z)/(4xz^2))^5 -> ((-2x^3z)/(4xz^2))^5`

Next simplification is to reduce the constants:

`((-2x^3z)/(4xz^2))^5 -> ((-2x^3z)/((color(red)(2*2))xz^2))^5 ->`

`((-color(red)(cancel(color(black)(2)))x^3z)/((color(red)(cancel(2)*2))xz^2))^5 -> ((-x^3z)/(2xz^2))^5`

Next, we can take advantage of two other rules for exponents:

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

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

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

Now, we can use yet another rule of exponents to further simplify this expression:

`color(red)((x^a)^b) = x^(a*b)`

`((-x^2)/(2z^1))^5 -> (-x^(color(red)(2*5)))/(2^color(red)(5)z^color(red)(1*5)) -> (-x^10)/(32z^5)`