To determine which expression is equivalent to the given expression [tex]\(\frac{12 x^9 y^4}{6 x^3 y^2}\)[/tex], let's simplify it step by step:
1. Simplify the coefficients:
The numerical part of the expression is [tex]\(\frac{12}{6}\)[/tex].
- [tex]\(\frac{12}{6} = 2\)[/tex]
2. Simplify the [tex]\(x\)[/tex] terms:
The [tex]\(x\)[/tex] part of the expression is [tex]\(\frac{x^9}{x^3}\)[/tex].
According to the properties of exponents, [tex]\(\frac{x^a}{x^b} = x^{a-b}\)[/tex].
- So, [tex]\(\frac{x^9}{x^3} = x^{9-3} = x^6\)[/tex]
3. Simplify the [tex]\(y\)[/tex] terms:
The [tex]\(y\)[/tex] part of the expression is [tex]\(\frac{y^4}{y^2}\)[/tex].
Using the same exponent rule:
- [tex]\(\frac{y^4}{y^2} = y^{4-2} = y^2\)[/tex]
Putting all these simplified parts together, we get:
[tex]\[
\frac{12 x^9 y^4}{6 x^3 y^2} = 2 x^6 y^2
\][/tex]
Therefore, the expression equivalent to [tex]\(\frac{12 x^9 y^4}{6 x^3 y^2}\)[/tex] is:
[tex]\[
\boxed{2 x^6 y^2}
\][/tex]
This matches option D, so the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
