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In the expansion of \((\sqrt[5]{3}+\sqrt[3]{2})^{15}\)
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Sum of all irrational terms is greater than the sum of all rational terms
Given binomial is \((\sqrt[5]{3}+\sqrt[3]{2})^{15}\)
\(\because\) The general term \(T_{r+1}={ }^{15} C_r 3^{\frac{15-r}{5}} 2^{\frac{r}{3}}\)
\(={ }^{15} C_r 3^{3-r / 5} 2^{1 / 3}\)
For rational terms \(r\) must be multiple of 15 , so possible values of \(r=0\) and \(15 \quad(\because 0 \leq r \leq 15)\)
\(\therefore\) Sum of rational terms \(={ }^{15} C_0 3^3+{ }^{15} C_{15} 2^5\)
\(=27+32=59.\)
\(\therefore\) The sum of all irrational terms is greater than the sum of all rational terms.
\(\because\) The general term \(T_{r+1}={ }^{15} C_r 3^{\frac{15-r}{5}} 2^{\frac{r}{3}}\)
\(={ }^{15} C_r 3^{3-r / 5} 2^{1 / 3}\)
For rational terms \(r\) must be multiple of 15 , so possible values of \(r=0\) and \(15 \quad(\because 0 \leq r \leq 15)\)
\(\therefore\) Sum of rational terms \(={ }^{15} C_0 3^3+{ }^{15} C_{15} 2^5\)
\(=27+32=59.\)
\(\therefore\) The sum of all irrational terms is greater than the sum of all rational terms.
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