Certainly! Let's simplify the given expression:
[tex]\[
\frac{\left(3 x^2 y^3\right)^6}{9 x y^2}
\][/tex]
First, we’ll simplify the numerator by raising the entire term [tex]\((3 x^2 y^3)\)[/tex] to the power of 6.
### Step 1: Simplify the numerator
We need to apply the exponent of 6 to each component inside the parentheses:
[tex]\[
(3 x^2 y^3)^6 = 3^6 \cdot (x^2)^6 \cdot (y^3)^6
\][/tex]
Calculating each term individually:
[tex]\[
3^6 = 729
\][/tex]
[tex]\[
(x^2)^6 = x^{2 \cdot 6} = x^{12}
\][/tex]
[tex]\[
(y^3)^6 = y^{3 \cdot 6} = y^{18}
\][/tex]
Combining these results, the simplified numerator is:
[tex]\[
729 x^{12} y^{18}
\][/tex]
### Step 2: Simplify the denominator
Now we simplify the denominator:
[tex]\[
9 x y^2
\][/tex]
### Step 3: Divide the numerator by the denominator
We substitute the simplified forms of the numerator and denominator into the original expression:
[tex]\[
\frac{729 x^{12} y^{18}}{9 x y^2}
\][/tex]
We can simplify this fraction by dividing the coefficients and subtracting the exponents of like bases:
[tex]\[
\frac{729 x^{12} y^{18}}{9 x y^2} = \frac{729}{9} \cdot \frac{x^{12}}{x^1} \cdot \frac{y^{18}}{y^2}
\][/tex]
Perform the division for each part:
[tex]\[
\frac{729}{9} = 81
\][/tex]
[tex]\[
\frac{x^{12}}{x^1} = x^{12-1} = x^{11}
\][/tex]
[tex]\[
\frac{y^{18}}{y^2} = y^{18-2} = y^{16}
\][/tex]
Putting all these together, we get:
[tex]\[
81 x^{11} y^{16}
\][/tex]
So the simplified expression is:
[tex]\[
81 x^{11} y^{16}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{81 x^{11} y^{16}}
\][/tex]
