The term independent of x in the expansion of $(x^{\frac{1}{3}}+\frac{1}{2x^{1/3}})^{18}, x0$ is:

Question

The term independent of x in the expansion of $(x^{\frac{1}{3}}+\frac{1}{2x^{1/3}})^{18}, x>0$ is:

Accepted Answer

To determine the term independent of $ x $ in the expansion of $(x^{\frac{1}{3}}+\frac{1}{2x^{1/3}})^{18}$, we need to find the term where the power of $ x $ is zero.

The general term in the binomial expansion of $(x^{\frac{1}{3}}+\frac{1}{2x^{1/3}})^{18}$ is given by:

$$
T_k = \binom{18}{k} \left(x^{\frac{1}{3}}\right)^{18-k} \left(\frac{1}{2x^{1/3}}\right)^k
$$

Simplifying the expression, we have:

$$
T_k = \binom{18}{k} \cdot x^{\frac{18-k}{3}} \cdot \frac{1}{2^k} \cdot x^{-\frac{k}{3}}
$$

$$
= \binom{18}{k} \cdot \frac{1}{2^k} \cdot x^{\frac{18-k-k}{3}}
$$

$$
= \binom{18}{k} \cdot \frac{1}{2^k} \cdot x^{\frac{18-2k}{3}}
$$

For the term to be independent of $ x $, the exponent of $ x $ must be zero:

$$
\frac{18-2k}{3} = 0
$$

Solving for $ k $:

$$
18 - 2k = 0
$$

$$
2k = 18
$$

$$
k = 9
$$

Thus, the term independent of $ x $ is when $ k = 9 $. Substituting $ k = 9 $ into the expression for the term, we get:

$$
T_9 = \binom{18}{9} \cdot \frac{1}{2^9}
$$

This matches Option 1: ${}^{18}C_{9}\cdot\frac{1}{2^{9}}$.

Now, let's explain why the other options are incorrect:

- **Option 2: ${}^{18}C_{7}\cdot\frac{1}{2^{7}}$**: This option corresponds to $ k = 7 $. Substituting $ k = 7 $ into the expression for the exponent of $ x $, we get $\frac{18-2(7)}{3} = \frac{4}{3}$, which is not zero. Therefore, this term is not independent of $ x $.

- **Option 3: ${}^{18}C_{5}\cdot\frac{1}{2^{5}}$**: This option corresponds to $ k = 5 $. Substituting $ k = 5 $ into the expression for the exponent of $ x $, we get $\frac{18-2(5)}{3} = \frac{8}{3}$, which is not zero. Therefore, this term is not independent of $ x $.

- **Option 4: ${}^{18}C_{3}\cdot\frac{1}{2^{3}}$**: This option corresponds to $ k = 3 $. Substituting $ k = 3 $ into the expression for the exponent of $ x $, we get $\frac{18-2(3)}{3} = \frac{12}{3} = 4$, which is not zero. Therefore, this term is not independent of $ x $.

In conclusion, Option 1 is the correct answer because it is the only option where the term is independent of $ x $.