Question

How do you simplify `(y^4/x^2)/((xy^2)/(2x^2))`?

Answer 1

`(2y^2)/x`

Explanation:

A complex fraction such as `(a/b)/(c/d)` can be written in a more familiar way:

`(a/b)/(c/d)` means the same as `(a/b) div (c/d)`

To divide by a fraction is the same as multiplying by its reciprocal.

`a/b color(limegreen)(div c/d) = a/b color(limegreen)(xx d/c) = (ad)/(bc)`

However if you compare this result to the original fraction you will see that we can simplify a complex fraction immediately:

`(color(red)(a)/color(blue)(b))/(color(blue)(c)/color(red)(d)) = color(red)(ad)/(color(blue)(bc))`
Applying this to the complex fraction given gives us:

`(color(red)(y^4)/color(blue)(x^2))/(color(blue)(xy^2)/color(red)(2x^2)) = (color(red)(2x^2 xx y^4))/(color(blue)(x^2 xx xy^2))" "` which now simplifies to:

`=(2y^2)/x`

Answer 2

`(2y^2)/x`

Explanation:

`(y^4/x^2)/((xy^2)/(2x^2))`

`:.y^4/x^2 xx (2x^2)/(xy^2`

`:.a^color(red)m*a^color(blue)n=a^(color(red)m+color(blue)n)`

`:.(y^color(red)(4-2) xx 2x^color(red)(2-2))/x^color(red)1`

`:.y^color(red)(4-2) xx 2x^color(red)(2-2-1)`

`:.y^color(red)2 xx 2x^color(red)(color(red)-1`

`:.y^color(red)2 xx 2 1/x^color(red)1`

`:.(2y^2)/x`