To simplify the given expression [tex]\(\frac{12 x^9 y^4}{6 x^3 y^2}\)[/tex], we will work through it step-by-step:
1. Simplify the coefficients:
- The coefficient in the numerator is [tex]\(12\)[/tex].
- The coefficient in the denominator is [tex]\(6\)[/tex].
- Divide the numerator coefficient by the denominator coefficient: [tex]\( \frac{12}{6} = 2 \)[/tex].
2. Simplify the exponents of [tex]\(x\)[/tex]:
- The exponent of [tex]\(x\)[/tex] in the numerator is [tex]\(9\)[/tex].
- The exponent of [tex]\(x\)[/tex] in the denominator is [tex]\(3\)[/tex].
- Subtract the exponent in the denominator from the exponent in the numerator: [tex]\(9 - 3 = 6\)[/tex].
- Thus, the [tex]\(x\)[/tex] term simplifies to [tex]\(x^6\)[/tex].
3. Simplify the exponents of [tex]\(y\)[/tex]:
- The exponent of [tex]\(y\)[/tex] in the numerator is [tex]\(4\)[/tex].
- The exponent of [tex]\(y\)[/tex] in the denominator is [tex]\(2\)[/tex].
- Subtract the exponent in the denominator from the exponent in the numerator: [tex]\(4 - 2 = 2\)[/tex].
- Thus, the [tex]\(y\)[/tex] term simplifies to [tex]\(y^2\)[/tex].
Combining these simplifications, the expression simplifies to:
[tex]\[ 2x^6y^2 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{2x^6y^2} \][/tex]
Thus, the correct option is:
D. [tex]\(2 x^6 y^2\)[/tex].
