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: $(a+b)^n = \sum_{r=0}^{n} {}^{n}{C_r} a^{n-r} b^r$ The $(r+1)^{th}$ term in the expansion is $T_{r+1} = {}^{n}{C_r} a^{n-r} b^r$. For a term to be rational, all its components must be rational. The binomial coefficient ${}^{n}{C_r}$ is always an integer (hence rational). Therefore, for $T_{r+1}$ to be rational, the terms $a^{n-r}$ and $b^r$ must result in rational numbers. This usually implies that if $a$ and $b$ involve roots (e.g., $a = x^{1/p}$, $b = y^{1/q}$), then their respective exponents $(n-r)$ and $r$ must be such that they eliminate the fractional powers, making the resulting values integers or simple fractions. In other words, the powers of prime factors in $a^{n-r}$ and $b^r$ must be integers.

Step-by-Step Solution

1. Identify the general term of the expansion. The given binomial expression is ${\left( {{4^{{1 \over 4}}} + {5^{{1 \over 6}}}} \right)^{120}}$. First, simplify the bases to their prime factors with fractional exponents: ${4^{{1 \over 4}}} = {(2^2)^{{1 \over 4}}} = {2^{{2 \over 4}}} = {2^{{1 \over 2}}}$ So, the expression becomes ${\left( {{2^{{1 \over 2}}} + {5^{{1 \over 6}}}} \right)^{120}}$. Using the binomial theorem, the general term, $T_{r+1}$, is: ${T_{r+1}} = {}^{120}{C_r} \left( {2^{{1 \over 2}}} \right)^{120-r} \left( {5^{{1 \over 6}}} \right)^{r}$ This step is crucial as it sets up the exponents we need to analyze for rationality.

2. Simplify the exponents in the general term. Apply the exponent rule $(x^a)^b = x^{ab}$: ${T_{r+1}} = {}^{120}{C_r} {2^{\frac{1}{2}(120-r)}} {5^{\frac{1}{6}r}}$ ${T_{r+1}} = {}^{120}{C_r} {2^{\frac{120-r}{2}}} {5^{\frac{r}{6}}}$ We simplify the exponents to clearly see the conditions for them to be integers.

3. Determine the conditions for the term to be rational. For $T_{r+1}$ to be a rational term, the powers of the prime bases (2 and 5) must be integers.

• For the term ${2^{\frac{120-r}{2}}}$ to be rational, the exponent $\frac{120-r}{2}$ must be an integer. This implies that $(120-r)$ must be an even number. If $120-r$ is even, then $r$ must also be an even number (since $120$ is even, and even - even = even).
• For the term ${5^{\frac{r}{6}}}$ to be rational, the exponent $\frac{r}{6}$ must be an integer. This implies that $r$ must be a multiple of 6.

4. Combine the conditions for $r$. From the above, $r$ must satisfy two conditions:

1. $r$ is an even number.
2. $r$ is a multiple of 6. If $r$ is a multiple of 6, it can be written as $6k$ for some integer $k$. Since $6k$ is always an even number, the first condition (r must be even) is automatically satisfied if $r$ is a multiple of 6. Therefore, for $T_{r+1}$ to be rational, $r$ must be a multiple of 6.

Tip: A common mistake is to consider the conditions separately without finding the least common multiple (LCM) of the denominators. Here, the denominators of the exponents are 2 and 6. The LCM of 2 and 6 is 6. So $r$ must be a multiple of 6. If the exponents were, for example, $\frac{120-r}{3}$ and $\frac{r}{5}$, then $120-r$ must be a multiple of 3 and $r$ must be a multiple of 5. This would mean $r$ is a multiple of 5 and $r$ must be such that $120-r$ is a multiple of 3.

5. Determine the possible values of $r$. In the binomial expansion of $(a+b)^n$, the value of $r$ ranges from $0$ to $n$ inclusive. In this case, $n=120$, so $0 \le r \le 120$. We need to find all multiples of 6 within this range: $r = \{0, 6, 12, 18, \dots, 120\}$ To find the number of such terms, let $r = 6k$, where $k$ is an integer. $0 \le 6k \le 120$ Divide by 6: $0 \le k \le 20$ The possible integer values for $k$ are $0, 1, 2, \dots, 20$.

6. Count the total number of rational terms. The number of values $k$ can take is $(20 - 0) + 1 = 21$. Each value of $k$ corresponds to a unique value of $r$, which in turn corresponds to a unique rational term in the expansion.

Therefore, there are 21 rational terms in the expansion.

Summary and Key Takeaway

To find the number of rational terms in a binomial expansion involving fractional powers:

1. Simplify the bases to their prime factor form, if possible, maintaining fractional exponents.
2. Write down the general term ($T_{r+1}$) using the binomial theorem.
3. Identify the exponents of any terms with fractional powers. For the term to be rational, these exponents must result in integers.
4. Determine the conditions on $r$ (the index of the term) such that all fractional exponents become integers. This often involves finding the Least Common Multiple (LCM) of the denominators of the fractional exponents.
5. Consider the valid range of $r$ (from $0$ to $n$).
6. Count the number of possible values of $r$ that satisfy all conditions within the given range.

The final answer is $\boxed{\text{21}}$.