Question

Which among the following terms in the expansion of ${(\sqrt[3]{3\frac{a}{\sqrt{b}}}+\sqrt{\frac{b}{\sqrt[3]{3a}}})}^{21}$will have the same powers for $a$ and $b$?

• A. $8^{\text{th}}$ term
• B. $9^{\text{th}}$ term
• C. ${10}^{\text{th}}$ term
• D. ${11}^{\text{th}}$ term

Answer 1

The correct option is C ${10}^{\text{th}}$ term

Let $(r+1{)}^{\text{th}}$ term has same power of $a$ and $b.$
$T_{r+1}={}^{21}{C}_r{\left(\sqrt[3]{3\frac{a}{\sqrt{b}}}\right)}^{21-r}{\left(\sqrt{\frac{b}{\sqrt[3]{3a}}}\right)}^r$
Exponent of $a$ in the given expansion $=\frac{21-r}{3}-\frac{r}{6}$
Exponent of $b$ in the given expansion $=\frac{r}{2}-\frac{21-r}{6}$
The exponents of $a$ and $b$ are the same, therefore,
$\frac{21-r}{3}-\frac{r}{6}=\frac{r}{2}-\frac{21-r}{6}$
On solving above equation, we get
$r=9$
$\Rightarrow {10}^{\text{th}}$ term has the same power of $a$ and $b.$