Key Concept: Binomial Theorem and Rational Terms

The binomial theorem states that for any real numbers $a$ and $b$, and any non-negative integer $n$, the expansion of $(a+b)^n$ is given by: $(a+b)^n = \sum_{r=0}^{n} C(n, r) a^{n-r} b^r$ where $C(n, r)$ (or $^nC_r$) is the binomial coefficient, $C(n, r) = \frac{n!}{r!(n-r)!}$.

The general term, also known as the $(r+1)^{th}$ term, in the expansion of $(a+b)^n$ is denoted by $T_{r+1}$ and is given by: $T_{r+1} = C(n, r) a^{n-r} b^r$ For a term in a binomial expansion involving surds (roots) to be rational, all the radical signs must disappear. This typically means that the exponents of the prime factors within the terms must be integers.

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Step-by-Step Solution

1. Identify the General Term ($T_{r+1}$) of the Expansion

The given binomial expression is $(7^{1/5} - 3^{1/10})^{60}$. Here, $a = 7^{1/5}$, $b = -3^{1/10}$, and $n = 60$.

Using the formula for the general term $T_{r+1} = C(n, r) a^{n-r} b^r$, we substitute the values: $T_{r+1} = C(60, r) (7^{1/5})^{60-r} (-3^{1/10})^r$

We expand the powers: $T_{r+1} = C(60, r) 7^{\frac{60-r}{5}} (-1)^r (3^{\frac{1}{10}})^r$ $T_{r+1} = C(60, r) (-1)^r 7^{\frac{60-r}{5}} 3^{\frac{r}{10}}$ Explanation: We express the general term to analyze its structure. The $(-1)^r$ factor determines the sign of the term but does not affect its rationality. The crucial parts for rationality are the exponents of the prime bases, 7 and 3.

2. Establish Conditions for a Term to be Rational

For $T_{r+1}$ to be a rational term, the powers of the prime numbers 7 and 3 must both be non-negative integers. This means: (i) The exponent of $7$, which is $\frac{60-r}{5}$, must be an integer. (ii) The exponent of $3$, which is $\frac{r}{10}$, must be an integer.

Explanation: A number involving roots (like $7^{1/5}$) is irrational. For the term to be rational, these roots must effectively cancel out, which happens when their exponents become integers. For example, $7^{2/5}$ is irrational, but $7^{5/5} = 7^1 = 7$ is rational.

3. Determine Possible Values of $r$ from the Conditions

From condition (i): $\frac{60-r}{5}$ must be an integer. This implies that $(60-r)$ must be a multiple of 5. Since 60 is a multiple of 5, for $(60-r)$ to be a multiple of 5, $r$ must also be a multiple of 5. So, $r \in \{0, 5, 10, 15, \ldots, 60\}$.

From condition (ii): $\frac{r}{10}$ must be an integer. This implies that $r$ must be a multiple of 10. So, $r \in \{0, 10, 20, 30, \ldots, 60\}$.

For a term to be rational, both conditions must be satisfied simultaneously. Therefore, $r$ must be a common multiple of 5 and 10. The least common multiple (LCM) of 5 and 10 is 10. Thus, $r$ must be a multiple of 10.

Also, remember that $r$ is the index in the binomial expansion and can range from $0$ to $n$. In this case, $0 \le r \le 60$.

Combining these, the possible values of $r$ for which the terms are rational are: $r \in \{0, 10, 20, 30, 40, 50, 60\}$.

Explanation: We are looking for values of $r$ that satisfy both constraints. By finding the multiples of 5 and 10, and then their common multiples, we narrow down the possibilities for $r$ within the valid range $[0, 60]$.

4. Calculate the Number of Rational Terms

The values of $r$ that yield rational terms are $0, 10, 20, 30, 40, 50, 60$. Counting these values, we find there are $7$ rational terms.

5. Calculate the Total Number of Terms in the Expansion

For a binomial expansion of $(a+b)^n$, the total number of terms is $n+1$. In this case, $n = 60$, so the total number of terms is $60 + 1 = 61$.

Explanation: This is a standard property of binomial expansions. For an exponent $n$, the terms range from $T_1$ (when $r=0$) to $T_{n+1}$ (when $r=n$), giving a total of $n+1$ terms.

6. Calculate the Number of Irrational Terms

The total number of terms is the sum of rational terms and irrational terms. Number of irrational terms = Total number of terms - Number of rational terms Number of irrational terms = $61 - 7 = 54$.

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Tips and Common Mistakes to Avoid

• Forgetting $n+1$: A common mistake is to assume the total number of terms is $n$ instead of $n+1$. Always remember to add 1 to the power of the binomial.
• Incorrectly identifying $r$ range: The value of $r$ always starts from $0$ and goes up to $n$. Ensure your multiples of 5/10 fall within this range.
• Only checking one exponent: A term is rational only if all surds are eliminated. If there are multiple bases with fractional exponents, all those exponents must become integers for the term to be rational.
• Negative signs: The sign of the term (due to $(-1)^r$) does not affect whether the term is rational or irrational. Focus solely on the exponents of the bases.

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Summary and Key Takeaway

To find the number of irrational terms in a binomial expansion involving roots, first identify the general term. Then, set up conditions for the exponents of all radical bases to be integers to determine the values of $r$ that yield rational terms. Count these rational terms and subtract them from the total number of terms ($n+1$) in the expansion to find the number of irrational terms. In this problem, there are $54$ irrational terms in the expansion of $(7^{1/5} - 3^{1/10})^{60}$.