The term void of x in the expression (x-3×2)18 is ____.

# The term void of x in the expression ${\left(x-\frac{3}{{x}^{2}}\right)}^{18}$ is ____.

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### Solution:

In this question, we need to find the term void of x in the expression of ${\left(x-\frac{3}{{x}^{2}}\right)}^{18}$ which means we need to find the term which has no x. For this, we will first find the general term of the given expression and then we will use properties of exponents to simplify the general term. General term would be in terms of $r$. Then we will put the value of the power of $x$ as zero which will give us the value of $r$. Using this value of. In general terms, we will get our required term. General term $r+1$ of expansion of is given as ${T}_{r+1}={n}_{{C}_{r}}{\left(a\right)}^{n-r}\cdot {\left(b\right)}^{r}$.
Properties of exponents that we will use are:
$\left(i\right){x}^{-1}=\frac{1}{x}$
$\left(\mathit{ii}\right){x}^{m}×{x}^{n}={x}^{m+n}$
$\left(\mathit{iii}\right){\left(\frac{a}{b}\right)}^{n}=\frac{{a}^{n}}{{b}^{n}}$
$\left(\mathit{iv}\right)\left({a}^{n}\right)n={a}^{\mathit{mn}}$
The general term is,
=${n}_{{C}_{r}}{X}^{18-r}{\left(-3\right)}^{r}{x}^{-2r}$
=${n}_{{C}_{r}}{X}^{18-3r}{\left(-3\right)}^{r}$
For term void of x
18−3r=0
$⇒$r=6
Then the term is ${18}_{{C}_{6}}{\left(-3\right)}^{6}={18}_{{C}_{6}}{\left(3\right)}^{6}$
Hence, option 2 is correct.

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