To find the number of irrational terms in the expansion of \((3^{1/4} + 5^{1/8})^{60}\), we can follow these steps:
### Step 1: Understand the Expansion
The expression \((3^{1/4} + 5^{1/8})^{60}\) can be expanded using the binomial theorem. According to the binomial theorem, the expansion will contain \(n + 1\) terms, where \(n\) is the exponent. In this case, \(n = 60\), so there will be \(60 + 1 = 61\) terms in the expansion.
### Step 2: General Term of the Expansion
The general term of the expansion can be expressed as:
\[
T_r = \binom{60}{r} (3^{1/4})^{60 - r} (5^{1/8})^r = \binom{60}{r} 3^{(60 - r)/4} 5^{r/8}
\]
where \(r\) varies from \(0\) to \(60\).
### Step 3: Determine Conditions for Rationality
For the term \(T_r\) to be rational, both \(3^{(60 - r)/4}\) and \(5^{r/8}\) must be rational numbers.
1. **Condition for \(3^{(60 - r)/4}\)**:
- This term is rational if \((60 - r)/4\) is an integer, which implies \(60 - r\) must be divisible by \(4\). Thus, \(r\) must satisfy:
\[
r \equiv 0 \mod 4
\]
2. **Condition for \(5^{r/8}\)**:
- This term is rational if \(r/8\) is an integer, which implies \(r\) must be divisible by \(8\). Thus, \(r\) must satisfy:
\[
r \equiv 0 \mod 8
\]
### Step 4: Find Values of \(r\)
Now, we need to find the values of \(r\) that satisfy both conditions:
- \(r \equiv 0 \mod 4\) means \(r\) can be \(0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60\).
- \(r \equiv 0 \mod 8\) means \(r\) can be \(0, 8, 16, 24, 32, 40, 48, 56\).
The common values of \(r\) that satisfy both conditions are:
- \(r = 0, 8, 16, 24, 32, 40, 48, 56\).
### Step 5: Count Rational Terms
The values of \(r\) that yield rational terms are \(0, 8, 16, 24, 32, 40, 48, 56\), which gives us a total of \(8\) rational terms.
### Step 6: Calculate Irrational Terms
Since there are \(61\) total terms in the expansion, the number of irrational terms \(n\) is given by:
\[
n = 61 - \text{(number of rational terms)} = 61 - 8 = 53
\]
### Final Answer
Thus, the value of \(n\) is \(53\).
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