Detailed Solution for Finding Irrational Terms in Binomial Expansion

Key Concept: The General Term of a Binomial Expansion For a binomial expansion of the form $(a+b)^N$, the general term (or $(r+1)^{th}$ term) is given by the formula: $T_{r+1} = {}^N C_r a^{N-r} b^r$ where $N$ is the power of the binomial, $r$ is the index of the term (starting from $r=0$ for the first term), and ${}^N C_r = \frac{N!}{r!(N-r)!}$ is the binomial coefficient. A term $T_{r+1}$ in the expansion is considered rational if, after simplification, it does not contain any radical expressions (like square roots, cube roots, etc.) whose values are irrational numbers. This typically means that the exponents of any prime bases in the term must be integers.

Step 1: Express the General Term for the Given Expansion The given expansion is ${\left( {{3^{1/4}} + {5^{1/8}}} \right)^{60}}$. Here, $a = 3^{1/4}$, $b = 5^{1/8}$, and $N = 60$. Substituting these into the general term formula, we get: $T_{r+1} = {}^{60}C_r {\left( {3^{1/4}} \right)^{60-r}} {\left( {5^{1/8}} \right)^r}$ To simplify the exponents, we use the property $(x^m)^n = x^{mn}$: $T_{r+1} = {}^{60}C_r \cdot 3^{\frac{60-r}{4}} \cdot 5^{\frac{r}{8}}$ Here, $r$ can take any integer value from $0$ to $60$, i.e., $r \in \{0, 1, 2, \ldots, 60\}$. This range covers all terms in the expansion.

Step 2: Establish Conditions for a Term to be Rational For the term $T_{r+1}$ to be rational, the powers of the prime bases (3 and 5) must be integers. This means:

1. The exponent of 3, which is $\frac{60-r}{4}$, must be a non-negative integer.
2. The exponent of 5, which is $\frac{r}{8}$, must be a non-negative integer.

Step 3: Determine the Values of 'r' that Yield Rational Terms

• Condition 1: $\frac{60-r}{4}$ must be an integer. This implies that $60-r$ must be a multiple of 4. Since 60 is a multiple of 4, for $60-r$ to be a multiple of 4, $r$ must also be a multiple of 4. So, possible values of $r$ are $0, 4, 8, 12, \ldots, 60$.

• Condition 2: $\frac{r}{8}$ must be an integer. This implies that $r$ must be a multiple of 8. So, possible values of $r$ are $0, 8, 16, 24, \ldots$.

• Combining Both Conditions: For $T_{r+1}$ to be rational, $r$ must satisfy both conditions. Therefore, $r$ must be a multiple of both 4 and 8. This means $r$ must be a multiple of the Least Common Multiple (LCM) of 4 and 8, which is 8. Considering the range of $r$ (from 0 to 60), the common values of $r$ that make the terms rational are: $r \in \{0, 8, 16, 24, 32, 40, 48, 56\}$.

Step 4: Count Rational and Irrational Terms

1. Number of Rational Terms: From the previous step, there are 8 values of $r$ that result in rational terms. Thus, there are 8 rational terms in the expansion.

2. Total Number of Terms: For an expansion $(a+b)^N$, the total number of terms is $N+1$. In this case, $N=60$, so the total number of terms is $60+1 = 61$.

3. Number of Irrational Terms (n): The number of irrational terms is the total number of terms minus the number of rational terms. $n = \text{Total terms} - \text{Rational terms}$ $n = 61 - 8 = 53$.

Step 5: Check Divisibility of $(n-1)$ The problem asks for $(n-1)$ to be divisible by which option. First, calculate $n-1$: $n-1 = 53 - 1 = 52$.

Now, we need to find which of the given options divides 52. Let's list the divisors of 52: Divisors of 52 are $1, 2, 4, 13, 26, 52$.

Let's check the given options: (A) 30: 52 is not divisible by 30. (B) 8: 52 is not divisible by 8. (C) 7: 52 is not divisible by 7. (D) 26: $52 \div 26 = 2$. Yes, 52 is divisible by 26.

Therefore, $(n-1)$ is divisible by 26.

Tips for Success & Common Mistakes:

• Don't forget $r=0$: Always include $r=0$ as it represents the first term, which can often be rational.
• Range of $r$: Ensure the values of $r$ found are within the valid range $[0, N]$.
• LCM vs. GCD: When combining conditions for exponents to be integers, you usually look for multiples of the LCM of the denominators.
• Total terms: Remember that for $(a+b)^N$, there are $N+1$ terms, not just $N$. This is a very common oversight.
• Careful with calculation: Double-check the arithmetic, especially when counting terms in sequences (e.g., using $\frac{\text{last term} - \text{first term}}{\text{common difference}} + 1$).

Summary and Key Takeaway To determine the number of irrational terms in a binomial expansion, first identify the general term. Then, set up conditions for the exponents of the prime bases to be integers, as this defines rational terms. Find the values of $r$ that satisfy these conditions. The count of such $r$ values gives the number of rational terms. Subtract this from the total number of terms ($N+1$) to find the number of irrational terms. Finally, perform the required calculation and check divisibility. In this problem, $n=53$, and $n-1=52$, which is divisible by 26.