Certainly! Let's simplify the given expression step by step.
Given the expression:
[tex]\[
\frac{18 x^4 - 8 x^2 y^2}{12 x^4 y - 16 x^3 y^2}
\][/tex]
### Step 1: Factor the Numerator
First, consider the numerator:
[tex]\[
18 x^4 - 8 x^2 y^2
\][/tex]
We can factor out the greatest common factor (GCF):
[tex]\[
= 2(9 x^4 - 4 x^2 y^2)
\][/tex]
Notice that [tex]\(9 x^4 - 4 x^2 y^2\)[/tex] is a difference of squares:
[tex]\[
9 x^4 - 4 x^2 y^2 = (3 x^2)^2 - (2 x y)^2
\][/tex]
We apply the difference of squares formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]:
[tex]\[
= 2 \left((3 x^2 - 2 y^2)(3 x^2 + 2 y^2)\right)
\][/tex]
So the numerator simplifies to:
[tex]\[
2(3 x^2 - 2 y^2)(3 x^2 + 2 y^2)
\][/tex]
### Step 2: Factor the Denominator
Next, consider the denominator:
[tex]\[
12 x^4 y - 16 x^3 y^2
\][/tex]
We can also factor out the GCF:
[tex]\[
= 4 x^3 y (3 x - 4 y)
\][/tex]
So the denominator simplifies to:
[tex]\[
4 x^3 y (3 x - 4 y)
\][/tex]
### Step 3: Simplify the Fraction
Now we have:
[tex]\[
\frac{2 (3 x^2 - 2 y^2) (3 x^2 + 2 y^2)}{4 x^3 y (3 x - 4 y)}
\][/tex]
We can see that [tex]\(2\)[/tex] in the numerator and [tex]\(4\)[/tex] in the denominator can be simplified:
[tex]\[
= \frac{(3 x^2 - 2 y^2)(3 x^2 + 2 y^2)}{2 x^3 y (3 x - 4 y)}
\][/tex]
The final simplified form of the given expression is:
[tex]\[
\frac{(3 x^2 - 2 y^2)(3 x^2 + 2 y^2)}{2 x y (3 x - 4 y)}
\][/tex]
In a concise form, it looks like:
[tex]\[
\frac{9x^2 - 4y^2}{2xy (3x - 4y)}
\][/tex]
Hence, the simplified expression is:
[tex]\[
\boxed{\frac{(9 x^2 - 4 y^2)}{2 x y (3 x - 4 y)}}
\][/tex]
