Introduction: Key Concept - Binomial Theorem

The Binomial Theorem provides a powerful method for expanding expressions of the form $(a+b)^n$. The general term, or $(r+1)^{th}$ term, in the expansion of $(a+b)^n$ is given by the formula:

$T_{r+1} = {}^{n}C_r a^{n-r} b^r$

where:

• $n$ is the power to which the binomial is raised.
• $r$ is an integer representing the term index, ranging from $0$ to $n$ (i.e., $0 \le r \le n$).
• ${}^{n}C_r$ is the binomial coefficient, calculated as $\frac{n!}{r!(n-r)!}$.

This formula is fundamental because it allows us to find any specific term in the expansion without having to compute all preceding terms.

Applying the Binomial Theorem to the Given Expression

Our given expression is $\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}$. By comparing this with the standard binomial form $(a+b)^n$, we can identify the components:

• $a = 2^{\frac{1}{5}}$
• $b = 5^{\frac{1}{3}}$
• $n = 15$

Now, we substitute these values into the general term formula:

$T_{r+1} = {}^{15}C_r \left(2^{\frac{1}{5}}\right)^{15-r} \left(5^{\frac{1}{3}}\right)^r$

Next, we simplify the exponents using the power rule $(x^m)^p = x^{mp}$:

$T_{r+1} = {}^{15}C_r 2^{\frac{15-r}{5}} 5^{\frac{r}{3}}$

This simplified expression represents any term in the expansion, where $r$ can be any integer from $0$ to $15$ (i.e., $r \in \{0, 1, 2, \dots, 15\}$).

Condition for Rational Terms

For a term $T_{r+1}$ to be rational, all its components must be rational numbers. The binomial coefficient ${}^{15}C_r$ is always an integer and thus rational. Therefore, the rationality of $T_{r+1}$ depends entirely on the terms involving the bases $2$ and $5$.

For $2^{\frac{15-r}{5}}$ and $5^{\frac{r}{3}}$ to be rational, their exponents must be non-negative integers. If an exponent is a fraction, the term would be an irrational root.

1. For the term involving $2$ ($2^{\frac{15-r}{5}}$): The exponent $\frac{15-r}{5}$ must be an integer. This means that $(15-r)$ must be perfectly divisible by $5$. Since $15$ is itself divisible by $5$, for $(15-r)$ to be divisible by $5$, $r$ must also be divisible by $5$. So, $r$ must be a multiple of $5$.

2. For the term involving $5$ ($5^{\frac{r}{3}}$): The exponent $\frac{r}{3}$ must be an integer. This means that $r$ must be perfectly divisible by $3$. So, $r$ must be a multiple of $3$.

Combining the Conditions: For $T_{r+1}$ to be a rational term, $r$ must satisfy both conditions simultaneously: $r$ must be a multiple of $5$ AND $r$ must be a multiple of $3$. This implies that $r$ must be a multiple of the Least Common Multiple (LCM) of $3$ and $5$.

LCM$(3, 5) = 15$.

Therefore, $r$ must be a multiple of $15$.

Determining Possible Values of $r$: We know that $r$ must be an integer and $0 \le r \le 15$. The only multiples of $15$ within this range are:

• $r=0$
• $r=15$

These are the only two values of $r$ that will yield rational terms in the expansion.

Tip for Avoiding Common Mistakes: Always remember to check the valid range for $r$ ($0 \le r \le n$). Overlooking this can lead to incorrect values of $r$ being included or excluded. Also, ensure that both exponents are integers, not just one.

Calculating the Rational Terms

Now, we substitute the valid values of $r$ back into the general term formula $T_{r+1} = {}^{15}C_r 2^{\frac{15-r}{5}} 5^{\frac{r}{3}}$ to find the actual rational terms.

1. For $r=0$: Substitute $r=0$ into the formula: $T_{0+1} = T_1 = {}^{15}C_0 2^{\frac{15-0}{5}} 5^{\frac{0}{3}}$ We know that ${}^{15}C_0 = 1$, $2^{\frac{15}{5}} = 2^3 = 8$, and $5^{\frac{0}{3}} = 5^0 = 1$. Therefore, $T_1 = 1 \cdot 8 \cdot 1 = 8$

2. For $r=15$: Substitute $r=15$ into the formula: $T_{15+1} = T_{16} = {}^{15}C_{15} 2^{\frac{15-15}{5}} 5^{\frac{15}{3}}$ We know that ${}^{15}C_{15} = 1$, $2^{\frac{0}{5}} = 2^0 = 1$, and $5^{\frac{15}{3}} = 5^5 = 3125$. Therefore, $T_{16} = 1 \cdot 1 \cdot 3125 = 3125$

Sum of Rational Terms

The sum of all rational terms in the expansion is the sum of the terms we calculated for $r=0$ and $r=15$.

Sum $= T_1 + T_{16} = 8 + 3125 = 3133$

Summary and Key Takeaway

To find the sum of all rational terms in a binomial expansion of the form $(\sqrt[p]{X} + \sqrt[q]{Y})^n$, follow these steps:

1. Find the General Term: Use the Binomial Theorem to write the general term $T_{r+1} = {}^{n}C_r (X^{1/p})^{n-r} (Y^{1/q})^r$ and simplify the exponents.
2. Apply Rationality Condition: For the term to be rational, the exponents of the bases $X$ and $Y$ must both be non-negative integers. This often means finding values of $r$ that are multiples of the denominators of the fractional exponents.
3. Determine Valid $r$ Values: Find the Least Common Multiple (LCM) of the denominators involved in the exponents, and identify all multiples of this LCM that fall within the valid range of $r$ ($0 \le r \le n$).
4. Calculate Terms: Substitute each valid $r$ value back into the general term to calculate the specific rational terms.
5. Sum the Terms: Add all the calculated rational terms together to get the final sum.

Key Takeaway: The problem boils down to understanding that rational terms arise when the fractional exponents in the binomial expansion simplify to integers. This simplification is achieved when the index $r$ is a suitable multiple that clears the denominators of these fractional powers.