Animated Solution for Mathematics - Binomial Theorem: If the term independent of $x$ in the expansion of $(\frac{3}{2}{x}^2-\frac{1}{3x}{)}^9$ is $k$, then $18k$ is equal to :

To solve for the term independent of $x$ in the expansion of $(\frac{3}{2}{x}^2-\frac{1}{3x}{)}^9$, we utilize the General Term Formula for a binomial expansion $(a+b{)}^n$:

In this specific problem, we identify our parameters as $n=9$, $a=\frac{3}{2}{x}^2$, and $b=-\frac{1}{3x}$. Keeping the negative sign attached to the term $b$ is a critical step to ensure the final coefficient is correct.

We substitute these values into the general term formula to begin our simplification:

To isolate the variable $x$, we separate the constants from the powers of $x$:

Applying the laws of exponents to the $x$ terms, we combine them:

The term is independent of $x$ when the exponent of $x$ is equal to zero. We set the exponent expression to zero and solve for $r$:

Since $r=6$, we are looking for the $7^{th}$ term of the expansion.

Now, we substitute $r=6$ back into the constant portion of our general term expression to find the value $k$:

Using the symmetry property $\left(\genfrac{}{}{0ex}{}{9}{6}\right)=\left(\genfrac{}{}{0ex}{}{9}{3}\right)$, we calculate the binomial coefficient:

Substituting this back into the equation for $k$:

Simplifying the fractions, we find:

The problem asks for the value of $18k$. Therefore:

The final answer is 7.