Reduce each rational expression

x^3-4xy^2/3x^3-2x^2y-8xy^2

factor out x in numerator and denominator to cancel,, it then becomes:

(x^2-4y^2)/(3x^2-2xy-8y^2)
factor both numerator and denominator:
[(x-2y)(x+2y)]/[(x-2y)(3x+4y)]
cancel (x-2y), therefore:
(x+2y)/(3x+4y)

so there,, :)

To reduce a rational expression, we need to simplify or cancel out common factors in both the numerator and the denominator.

The given rational expression is:

(x^3 - 4xy^2) / (3x^3 - 2x^2y - 8xy^2)

First, let's factor the numerator and denominator separately:

Numerator:
x^3 - 4xy^2

We can factor out the common factor of x:
x(x^2 - 4y^2)

Now, let's factor the difference of squares in the parentheses:
x(x + 2y)(x - 2y)

Denominator:
3x^3 - 2x^2y - 8xy^2

There are no common factors in the denominator that can be factored out further.

Now that we have factored both the numerator and the denominator, we can cancel out any common factors:

(x(x + 2y)(x - 2y)) / (3x^3 - 2x^2y - 8xy^2)

And that's the reduced form of the given rational expression.