Question:

Step 1: Write down the general term in the binomial expansion. The general term, $T_{r+1}$, in the expansion of $(a+b)^n$ is given by $T_{r+1} = \binom{n}{r} a^{n-r} b^r$. For the given expansion $\left(2^{1/3}+5^{1/5}\right)^{15}$, we have $a=2^{1/3}$, $b=5^{1/5}$, and $n=15$. Substituting these values, the general term is: $$T_{r+1} = \binom{15}{r} \left(2^{1/3}\right)^{15-r} \left(5^{1/5}\right)^r$$ $$T_{r+1} = \binom{15}{r} 2^{\frac{15-r}{3}} 5^{\frac{r}{5}}$$ Step 2: Determine the conditions for a term to be rational. For $T_{r+1}$ to be a rational term, the exponents of 2 and 5 must be non-negative integers. This implies two conditions: 1. The exponent $\frac{15-r}{3}$ must be an integer. This requires $15-r$ to be a multiple of 3. Since 15 is a multiple of 3, $r$ must also be a multiple of 3. 2. The exponent $\frac{r}{5}$ must be an integer. This requires $r$ to be a multiple of 5. Step 3: Find the possible values of $r$. From the conditions in Step 2, $r$ must be a common multiple of 3 and 5. The least common multiple (LCM) of 3 and 5 is 15. In a binomial expansion $(a+b)^n$, the value of $r$ ranges from $0$ to $n$. Here, $0 \le r \le 15$. The values of $r$ in the range $[0, 15]$ that are multiples of both 3 and 5 are: * $r=0$ * $r=15$ Step 4: Calculate the first rational term (for $r=0$). For $r=0$, the term is $T_{0+1} = T_1$. $$T_1 = 8$$ Step 5: Calculate the second rational term (for $r=15$). For $r=15$, the term is $T_{15+1} = T_{16}$. $$T_{16} = 3125$$ Step 6: Calculate the sum of the rational terms. The sum of all rational terms is the sum of $T_1$ and $T_{16}$. $$\text{Sum} = T_1 + T_{16}$$ $$\text{Sum} = 8 + 3125$$ $$\text{Sum} = 3133$$ The sum of all rational terms in the expansion is $3133$.