Given $\left(\sqrt[4]{3} + \sqrt[4]{2}\right)^{10}$, find the term independent of $z$.
Step-by-Step Solution
Key Concept: For a term to be independent of a variable, the exponent of that variable must equal zero; set up and solve the exponent equation
The general term is $T_{r+1} = {}^{10}C_r\left(\sqrt[4]{3}\right)^{10-r}\left(\sqrt[4]{2}\right)^r = {}^{10}C_r(3)^{\frac{10-r}{4}}(2)^{\frac{r}{4}}$. For the term independent of $z$, we set $\frac{10-r}{4} - 2r = 0$, which gives $r = 2$. Thus $T_{2+1} = T_3 = {}^{10}C_2(3)^2 = 45 \times \frac{3^2}{1} = 45 \times \frac{9}{1} = 45$.
Correct Answer: 45