Question

Find the term independent of x in the expression of ${\left( {\sqrt {\dfrac{x}{3}} + \dfrac{{\sqrt 3 }}{{2{x^2}}}} \right)^{10}}$.

Answer 1

Hint: The term independent of x means that the term in which power of x= 0. We will suppose that ${\text{r+1}}^{\text{th}}$ term is the term independent of x in the given equation and then we will find r.

Complete step by step solution: Let (r+1) be the term independent of x.
We know that ${(r + 1)^{th}}$ term of a binomial expansion (a+b)n is given by the formula:
${T_{r + 1}} = \left( {}_r^n \right) C{(a)^{n - r}}{(b)^n}$
Here, n= 10, a= $\sqrt {\dfrac{x}{3}} $and b= $\dfrac{{\sqrt 3 }}{{2{x^2}}}$
Substituting these values in the equation of $\text{T}_{\text{r+1}}$, we get
$
   \Rightarrow {T_{r + 1}} = \left( {}_r^{10} \right) C{\left( {\sqrt {\dfrac{x}{3}} } \right)^{10 - r}}{\left( {\dfrac{{\sqrt 3 }}{{2{x^2}}}} \right)^r} \\
   \Rightarrow {T_{r + 1}} = \left( {}_r^{10} \right) C{\left( {\dfrac{x}{3}} \right)^{\dfrac{{10 - r}}{2}}}{3^{\dfrac{r}{2}}}\left( {\dfrac{1}{{{2^r}{x^{2r}}}}} \right) \\
   \Rightarrow {T_{r + 1}} = \left( {}_r^{10} \right) C{3^{\dfrac{r}{2} - \dfrac{{10 - r}}{2}}}{2^{ - r}}{x^{\dfrac{{10 - r}}{2} - 2r}} \\
 $
Here, we are looking for a term which is independent of x, so the power of x must be 0.
$
   \Rightarrow \dfrac{{10 - r}}{2} - 2r = 0 \\
   \Rightarrow 10 - r - 4r = 0 \\
   \Rightarrow 10 - 5r = 0 \\
   \Rightarrow r = 2 \\
 $
Now when we know that ${(r + 1)^{th}}$ term is independent and we have also calculated the value of r, then the term independent of x will be (r+1) = 2+1= 3.

Therefore, ${3}^{\text{rd}}$ term is independent of x in the expansion of ${\left( {\sqrt {\dfrac{x}{3}} + \dfrac{{\sqrt 3 }}{{2{x^2}}}} \right)^{10}}$.

Note: When we need to find a term which is independent of any variable, say, x, the n we just put the power of all terms containing x equal to zero. Further, if we need to find the value of the ${3}^{\text{rd}}$ term then we will just put the value of r in the equation of nth term in the binomial expansion of the given equation. You just need to take care of using the correct formula and make sure that you don’t miss any term containing x as it will lead to wrong results.