Certainly! Let's simplify the given algebraic expression step-by-step:
The expression is:
[tex]\[
\frac{18 x^4 - 8 x^2 y^2}{12 x^4 y - 16 x^3 y^2}
\][/tex]
### Step 1: Factorize the numerator and the denominator
First, let's look at the numerator, [tex]\(18 x^4 - 8 x^2 y^2\)[/tex]:
- Both terms [tex]\(18 x^4\)[/tex] and [tex]\(-8 x^2 y^2\)[/tex] have a common factor of [tex]\(2 x^2\)[/tex].
- Factor out [tex]\(2 x^2\)[/tex]:
[tex]\[
18 x^4 - 8 x^2 y^2 = 2 x^2 (9 x^2 - 4 y^2)
\][/tex]
Now, the denominator, [tex]\(12 x^4 y - 16 x^3 y^2\)[/tex]:
- Both terms [tex]\(12 x^4 y\)[/tex] and [tex]\(-16 x^3 y^2\)[/tex] have a common factor of [tex]\(4 x^3 y\)[/tex].
- Factor out [tex]\(4 x^3 y\)[/tex]:
[tex]\[
12 x^4 y - 16 x^3 y^2 = 4 x^3 y (3 x - 4 y)
\][/tex]
Notice that we need to simplify both expressions further if possible.
### Step 2: Recognize special factorizations
The numerator [tex]\(9 x^2 - 4 y^2\)[/tex] is a difference of squares:
[tex]\[
9 x^2 - 4 y^2 = (3x)^2 - (2y)^2 = (3x - 2y)(3x + 2y)
\][/tex]
### Step 3: Substitute factorized forms back into the expression:
Now substitute back into the original fraction:
[tex]\[
\frac{2 x^2 (3x - 2y)(3x + 2y)}{4 x^3 y (3x - 4y)}
\][/tex]
### Step 4: Simplify the fraction
- Firstly, cancel any common factors from the numerator and the denominator:
- Notice [tex]\(x^2\)[/tex] in the numerator cancels with [tex]\(x^3\)[/tex] in the denominator:
[tex]\[
\frac{2 (3x - 2y)(3x + 2y)}{4 x y (3x - 4y)} = \frac{(3x - 2y)(3x + 2y)}{2 x y (3x - 4y)}
\][/tex]
- There are no more common factors to cancel, so the fraction is in its simplest form now.
### Final Answer:
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{(3x - 2y)(3x + 2y)}{2 x y (3x - 4y)}}
\][/tex]
