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# The number of irrational terms in the expansion of ${{\left( {{4}^{\dfrac{1}{5}}}+{{7}^{\dfrac{1}{10}}} \right)}^{45}}$ is?(a) 40(b) 5(c) 41(d) None of these Verified
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Hint: Use the fact that ${{\left( x+y \right)}^{n}}$ has (n + 1) terms and try to find the number of rational terms in the expression ${{\left( {{4}^{\dfrac{1}{5}}}+{{7}^{\dfrac{1}{10}}} \right)}^{45}}$. Use the formula for the general term of the expression ${{\left( x+y \right)}^{n}}$ given as ${{T}_{r+1}}={}^{n}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}$ and simplify the general term of the expression ${{\left( {{4}^{\dfrac{1}{5}}}+{{7}^{\dfrac{1}{10}}} \right)}^{45}}$. Check for which values of r we will get the rational terms. Subtract the number of rational terms from the total number of terms to get the number of irrational terms.

Here we have been provided with the expression ${{\left( {{4}^{\dfrac{1}{5}}}+{{7}^{\dfrac{1}{10}}} \right)}^{45}}$ and we are asked number of irrational terms in its expansion. Here we will find the number of rational terms and then subtract it from the total number of terms to get the answer. Let us assume the expression as E, so we have,
$\Rightarrow E={{\left( {{4}^{\dfrac{1}{5}}}+{{7}^{\dfrac{1}{10}}} \right)}^{45}}$
We know that number of terms in the binomial expression ${{\left( x+y \right)}^{n}}$ is (n + 1), so the total number of terms in ${{\left( {{4}^{\dfrac{1}{5}}}+{{7}^{\dfrac{1}{10}}} \right)}^{45}}$ will be 46. Now, the general term of the expression ${{\left( x+y \right)}^{n}}$ is given as ${{T}_{r+1}}={}^{n}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}$, so replacing x with ${{4}^{\dfrac{1}{5}}}$ and y with ${{7}^{\dfrac{1}{10}}}$ we get the general term of ${{\left( {{4}^{\dfrac{1}{5}}}+{{7}^{\dfrac{1}{10}}} \right)}^{45}}$ as: -
\begin{align} & \Rightarrow {{T}_{r+1}}={}^{45}{{C}_{r}}{{\left( {{4}^{\dfrac{1}{5}}} \right)}^{45-r}}{{\left( {{7}^{\dfrac{1}{10}}} \right)}^{r}} \\ & \Rightarrow {{T}_{r+1}}={}^{45}{{C}_{r}}\left( {{4}^{\dfrac{45-r}{5}}} \right)\left( {{7}^{\dfrac{r}{10}}} \right) \\ & \Rightarrow {{T}_{r+1}}={}^{45}{{C}_{r}}\left( {{4}^{9-\dfrac{r}{5}}} \right)\left( {{7}^{\dfrac{r}{10}}} \right) \\ \end{align}
Using the formula of exponents given as ${{a}^{m-n}}=\dfrac{{{a}^{m}}}{{{a}^{n}}}$ we get,
$\Rightarrow {{T}_{r+1}}={}^{45}{{C}_{r}}\left( {{4}^{9}} \right)\left( \dfrac{{{7}^{\dfrac{r}{10}}}}{{{4}^{\dfrac{r}{5}}}} \right)$
$\Rightarrow$ Number of irrational terms = 46 – 5
$\Rightarrow$ Number of irrational terms = 41