<p>The number of rational terms in the expansion of \(\left(2^{1/5} + 3^{1/10}\right)^{45}\) is</p>
Step-by-Step Solution
Key Concept: A term in the expansion is rational only when both exponents of 2 and 3 are integers. Use the general term formula and set conditions on the binomial coefficient index so that fractional powers become whole numbers.
<p><strong>Step 1:</strong> Write the general term in the expansion of $(2^{1/5} + 3^{1/10})^{45}$:</p><p>$T_{r+1} = \binom{45}{r}(2^{1/5})^{45-r}(3^{1/10})^r = \binom{45}{r} \cdot 2^{(45-r)/5} \cdot 3^{r/10}$</p><p><strong>Step 2:</strong> For the term to be rational, both exponents must be integers:</p><p>• Exponent of 2: $\frac{45-r}{5}$ must be an integer → $45-r \equiv 0 \pmod{5}$ → $r \equiv 0 \pmod{5}$</p><p>• Exponent of 3: $\frac{r}{10}$ must be an integer → $r \equiv 0 \pmod{10}$</p><p><strong>Step 3:</strong> Find values of $r$ satisfying BOTH conditions simultaneously:</p><p>$r$ must be divisible by both 5 and 10 → $r \equiv 0 \pmod{10}$</p><p><strong>Step 4:</strong> Count valid values where $0 \leq r \leq 45$:</p><p>$r \in \{0, 10, 20, 30, 40\}$</p><p>∴ Number of rational terms = <strong>5</strong></p>
Correct Answer: C