Question

The sum of all rational terms in the expansion of \( \left( 1 + 2^{1/3} + 3^{1/2} \right)^6 \) is equal to                 

Hint:

To find the sum of rational terms in a multinomial expansion, ensure that the exponents of the irrational terms result in integers, and then apply the multinomial theorem to calculate the coefficients.

Answer 1

The provided expression is: \[\left( 1 + 2^{1/3} + 3^{1/2} \right)^6\] To determine the sum of all rational terms in its expansion, the multinomial theorem is applied. The multinomial coefficient for the expansion is calculated as: \[\frac{6!}{r_1! r_2! r_3!} (1)^{r_1} \left( 2^{1/3} \right)^{r_2} \left( 3^{1/2} \right)^{r_3}\] This simplifies to: \[\frac{6!}{r_1! r_2! r_3!} \times (1)^{r_1} \times \left( 2^{r_2/3} \right) \times \left( 3^{r_3/2} \right)\] Rational terms occur when the powers of \(2\) and \(3\) are integers, which necessitates \(r_2\) being a multiple of 3 and \(r_3\) being a multiple of 2. By substituting the appropriate values for \(r_1\), \(r_2\), and \(r_3\) into the multinomial expansion and calculating the terms, the rational term is found for: \[r_1 = 6, \quad r_2 = 0, \quad r_3 = 0\] The sum of the rational terms is calculated as: \[1 + 45 + 135 + 27 + 40 + 360 + 4 = 612\] Therefore, the sum of all rational terms is \( 612 \).