Question:
The sum of all rational terms in the expansion of \( \left( 1 + 2^{1/3} + 3^{1/2} \right)^6 \) is equal to
The sum of all rational terms in the expansion of \( \left( 1 + 2^{1/3} + 3^{1/2} \right)^6 \) is equal to
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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.
Updated On: Jan 14, 2026
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Approach Solution - 1
The given expression is:
\[
\left( 1 + 2^{1/3} + 3^{1/2} \right)^6
\]
To find the sum of all rational terms in the expansion, we use the multinomial theorem.
First, we calculate the multinomial coefficient for the expansion of the given expression, where the sum of the rational terms is expressed 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:
\[
\frac{6!}{r_1! r_2! r_3!} \times (1)^{r_1} \times \left( 2^{r_2/3} \right) \times \left( 3^{r_3/2} \right)
\]
Next, we calculate the rational terms. The rational terms are those where the powers of \(2\) and \(3\) are integers, meaning \(r_2\) must be a multiple of 3 and \(r_3\) must be a multiple of 2.
Substitute the corresponding values of \(r_1\), \(r_2\), and \(r_3\) into the multinomial expansion, and calculate the terms.
\[
r_1 = 6, \quad r_2 = 0, \quad r_3 = 0 \quad \text{for the rational term}
\]
Finally, we sum the rational terms to get the total:
\[
1 + 45 + 135 + 27 + 40 + 360 + 4 = 612
\]
Thus, the sum of all rational terms is \( 612 \).
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Approach Solution -2
sum of rational terms in \(\left(1+2^{1/3}+3^{1/2}\right)^6\)
Condition for a term to be rational: if the multinomial term \[ \frac{6!}{r_1!r_2!r_3!}\,1^{r_1}\,(2^{1/3})^{r_2}\,(3^{1/2})^{r_3} \] is rational, then the exponents of 2 and 3 must be integers. So \(r_2\) must be divisible by 3 and \(r_3\) must be even, with \(r_1+r_2+r_3=6\).
All solutions \((r_1,r_2,r_3)\) with these constraints:
- \((6,0,0)\): coefficient \(1\), value \(1\).
- \((4,0,2)\): coefficient \(\dfrac{6!}{4!0!2!}=15\), value \(15\cdot 3^{1}=45\).
- \((2,0,4)\): coefficient \(\dfrac{6!}{2!0!4!}=15\), value \(15\cdot 3^{2}=135\).
- \((0,0,6)\): coefficient \(1\), value \(3^{3}=27\).
- \((3,3,0)\): coefficient \(\dfrac{6!}{3!3!0!}=20\), value \(20\cdot 2^{1}=40\).
- \((1,3,2)\): coefficient \(\dfrac{6!}{1!3!2!}=60\), value \(60\cdot 2^{1}\cdot 3^{1}=360\).
- \((0,6,0)\): coefficient \(1\), value \(2^{2}=4\).
Sum of these rational terms:
\[ 1 + 45 + 135 + 27 + 40 + 360 + 4 = 612. \]
Final answer:
\(\boxed{612}\)
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