Could someone help me with this question
Simplify the rational expression.
(3x^2y+9xy^2-12y^3)/(36x^3y-27x^2y^2-9xy^3)
Factor the numerator and denominator:
3y(x^2 + 3xy - 4y^2)/9xy(4x^2-3xy-y^2),
Factor the numerator and denominator into 2 binomials:
3y(x - y)(x + 4y) / 9xy(x - y)(4x + y),
Divide the numerator and denominator by 3y:
(x - y)(x + 4y) / 3x(x - y)(4x + y),
Cancel the 2 (x - y) factors:
(x + 4y) / 3x(4x + y).
To simplify the rational expression, we can factor out any common terms in the numerator and denominator and then cancel out those common terms.
In the numerator, we have 3x^2y, 9xy^2, and -12y^3. We can factor out y from all of these terms:
y(3x^2 + 9xy - 12y^2)
In the denominator, we have 36x^3y, -27x^2y^2, and -9xy^3. We can factor out 9xy from all of these terms:
9xy(4x^2 - 3xy - y^2)
Now we can rewrite the rational expression factored:
y(3x^2 + 9xy - 12y^2) / (9xy)(4x^2 - 3xy - y^2)
Next, we can cancel out the common factors in the numerator and denominator, which is y and 9xy:
(3x^2 + 9xy - 12y^2) / (4x^2 - 3xy - y^2)
Therefore, the simplified form of the rational expression is (3x^2 + 9xy - 12y^2) / (4x^2 - 3xy - y^2).