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(A) The general term in the expansion of $(a+b)^n$ is given by $T_{r+1} = ^nC_r a^{n-r} b^r$.For the given expression $\left(x^{1/3} + \frac{1}{2x^{1/

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$^{18}C_9 \cdot 2^{-9}$

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(A) The general term in the expansion of $(a+b)^n$ is given by $T_{r+1} = ^nC_r a^{n-r} b^r$.For the given expression $\left(x^{1/3} + \frac{1}{2x^{1/3}}\right)^{18}$,we have:$T_{r+1} = ^{18}C_r (x^{1/3})^{18-r} \left(\frac{1}{2x^{1/3}}\right)^r$$T_{r+1} = ^{18}C_r \cdot x^{\frac{18-r}{3}} \cdot \frac{1}{2^r \cdot x^{r/3}}$$T_{r+1} = ^{18}C_r \cdot \frac{1}{2^r} \cdot x^{\frac{18-2r}{3}}$For the term to be independent of $x$,the exponent of $x$ must be $0$:$\frac{18-2r}{3} = 0 \implies 18-2r = 0 \implies r = 9$.Substituting $r=9$ into the expression,the independent term is:$T_{9+1} = ^{18}C_9 \cdot \frac{1}{2^9} = ^{18}C_9 \cdot 2^{-9}$.

Find the term independent of $x$ in the expansion of $\left(\sqrt[3]{x}+\frac{1}{2 \sqrt[3]{x}}\right)^{18}, x > 0$.

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