Question

How do you simplify `(\frac { - 3x ^ { 2} y } { 2x y ^ { - 2} } ) ^ { - 2} \times ( \frac { 9x ^ { 2} } { 4y ^ { 4} } )`?

Answer 1

See a solution process below:

Explanation:

First, use these rules of exponents to simplify the term on the left:

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

`((-3x^2y)/(2xy^-2))^-2 xx ((9x^2)/(4y^4)) =>`

`((-3^color(red)(1)x^color(red)(2)y^color(red)(1))/(2^color(red)(1)x^color(red)(1)y^color(red)(-2)))^color(blue)(-2) xx ((9x^2)/(4y^4)) =>`

`((-3^(color(red)(1)xxcolor(blue)(-2))x^(color(red)(2)xxcolor(blue)(-2))y^(color(red)(1)xxcolor(blue)(-2)))/(2^(color(red)(1)xxcolor(blue)(-2))x^(color(red)(1)xxcolor(blue)(-2))y^(color(red)(-2)xxcolor(blue)(-2)))) xx ((9x^2)/(4y^4)) =>`

`((-3^-2x^-4y^-2)/(2^-2x^-2y^4)) xx ((9x^2)/(4y^4))`

Next, rewrite the expression as:

`(-(3^-2 xx 9)/(2^-2 xx 4))((x^-4 xx x^2)/x^-2)(y^-2/(y^4 xx y^4))`

Next, use these rules of exponents to simplify the constants:

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

`(-(3^color(red)(-2) xx 9)/(2^color(blue)(-2) xx 4))((x^-4 xx x^2)/x^-2)(y^-2/(y^4 xx y^4)) =>`

`(-(2^color(blue)(- -2) xx 9)/(3^color(red)(- -2) xx 4))((x^-4 xx x^2)/x^-2)(y^-2/(y^4 xx y^4)) =>`

`(-(2^color(blue)(2) xx 9)/(3^color(red)(2) xx 4))((x^-4 xx x^2)/x^-2)(y^-2/(y^4 xx y^4)) =>`

`(-(4 xx 9)/(9 xx 4))((x^-4 xx x^2)/x^-2)(y^-2/(y^4 xx y^4)) =>`

`(-36/36)((x^-4 xx x^2)/x^-2)(y^-2/(y^4 xx y^4)) =>`

`-1((x^-4 xx x^2)/x^-2)(y^-2/(y^4 xx y^4)) =>`

`-((x^-4 xx x^2)/x^-2)(y^-2/(y^4 xx y^4))`

Then, use this rule of exponents to simplify the numerator of the `x` term:

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

`-((x^color(red)(-4) xx x^color(blue)(2))/x^-2)(y^-2/(y^4 xx y^4)) =>`

`-(x^(color(red)(-4)+color(blue)(2))/x^-2)(y^-2/(y^4 xx y^4)) =>`

`-(x^-2/x^-2)(y^-2/(y^4 xx y^4)) =>`

`-(1)(y^-2/(y^4 xx y^4)) =>`

`-(y^-2/(y^4 xx y^4))`

Now, use these rules of exponents to simplify the `y` terms:

`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))`

`-(y^-2/(y^color(red)(4) xx y^color(blue)(4))) =>`

`-(y^-2/y^(color(red)(4) + color(blue)(4))) =>`

`-(y^color(red)(-2)/(y^color(blue)(8))) =>`

`-(1/y^(color(blue)(8)-color(red)(-2))) =>`

`-(1/y^(color(blue)(8)+color(red)(2))) =>`

`-1/y^10`