Elaborate Solution for Integral Terms in Binomial Expansion

---

1. Key Concept: The General Term of a Binomial Expansion

The problem asks for the number of integral terms in the expansion of ${\left( {\sqrt 3 + \root 8 \of 5 } \right)^{256}}$. To find these terms, we first need to write down the general term of the binomial expansion.

For any binomial expansion of the form $(a+b)^n$, the general term (or the $(r+1)^{th}$ term), denoted as $T_{r+1}$, is given by the formula: $T_{r+1} = {}^{n}C_r \cdot a^{n-r} \cdot b^r$ where $r$ is an integer ranging from $0, 1, 2, \dots, n$.

---

2. Applying the Formula to the Given Expansion

In our problem, the given expansion is ${\left( {\sqrt 3 + \root 8 \of 5 } \right)^{256}}$. Here, we identify the components:

• $n = 256$
• $a = \sqrt{3} = 3^{1/2}$
• $b = \root 8 \of 5 = 5^{1/8}$

Substituting these into the general term formula, we get: $T_{r+1} = {}^{256}C_r \cdot \left(3^{1/2}\right)^{256-r} \cdot \left(5^{1/8}\right)^r$ Using the power rule $(x^m)^p = x^{mp}$, we simplify the exponents: $T_{r+1} = {}^{256}C_r \cdot 3^{\frac{256-r}{2}} \cdot 5^{\frac{r}{8}}$ This is the general form of any term in the expansion.

---

3. Conditions for an Integral Term

For a term $T_{r+1}$ to be an integral term (i.e., a whole number), two conditions must be satisfied:

• The binomial coefficient ${}^{256}C_r$ must be an integer. This is always true by definition, as ${}^{n}C_r = \frac{n!}{r!(n-r)!}$ always yields an integer for $0 \le r \le n$.
• The powers of the base numbers (3 and 5 in this case) must result in integers. This means that the exponents $\frac{256-r}{2}$ and $\frac{r}{8}$ must both be non-negative integers. If an exponent were a fraction (e.g., $3^{1/2}$), the term would contain a radical and thus not be an integer (unless the base itself was a perfect power that cancels the fractional exponent, which is not the case here for prime bases 3 and 5).

---

4. Deriving Divisibility Conditions for 'r'

Based on the requirement that the exponents must be non-negative integers, we establish two conditions for $r$:

Condition 1: For the exponent of 3 The exponent $\frac{256-r}{2}$ must be an integer. This implies that $(256-r)$ must be an even number. Since 256 is an even number, for $(256-r)$ to be even, $r$ must also be an even number. (Even - Even = Even). So, $r$ must be a multiple of 2.

Condition 2: For the exponent of 5 The exponent $\frac{r}{8}$ must be an integer. This implies that $r$ must be a multiple of 8.

---

5. Combining the Conditions and Determining the Range of 'r'

We need $r$ to satisfy both Condition 1 and Condition 2.

• $r$ must be a multiple of 2 (i.e., $r = 2k_1$ for some integer $k_1$).
• $r$ must be a multiple of 8 (i.e., $r = 8k_2$ for some integer $k_2$).

If $r$ is a multiple of 8, it is automatically a multiple of 2 (since $8 = 2 \times 4$). Therefore, the stronger condition, $r$ must be a multiple of 8, takes precedence.

Additionally, in a binomial expansion $(a+b)^n$, the index $r$ for the general term $T_{r+1}$ can only take integer values from $0$ up to $n$. In this problem, $n = 256$. So, the valid range for $r$ is $0 \le r \le 256$.

---

6. Listing Possible Values of 'r'

Combining the derived conditions, we need to find all integer values of $r$ such that:

1. $r$ is a multiple of 8.
2. $0 \le r \le 256$.

The values of $r$ that satisfy these conditions are: $0, 8, 16, 24, \dots, 256$.

---

7. Counting the Number of Integral Terms

The sequence of possible $r$ values forms an arithmetic progression (AP) with:

• First term ($a_1$) = 0
• Common difference ($d$) = 8
• Last term ($a_N$) = 256 (where $N$ is the number of terms)

We use the formula for the $N^{th}$ term of an AP: $a_N = a_1 + (N-1)d$. $256 = 0 + (N-1)8$ $256 = 8(N-1)$ To find $N$, divide both sides by 8: $\frac{256}{8} = N-1$ $32 = N-1$ $N = 32 + 1$ $N = 33$

Thus, there are 33 integral terms in the expansion.

---

8. Tips and Common Mistakes to Avoid

• Always check the range of 'r': Remember that $r$ must be between $0$ and $n$ (inclusive).
• Consider all exponents: For a term to be integral, all fractional exponents must simplify to integers. For example, if there were a $\sqrt[3]{7}$ term, its exponent would also need to be an integer.
• Simplify combined conditions: If 'r' must be a multiple of 'X' and 'Y', then 'r' must be a multiple of the Least Common Multiple (LCM) of X and Y. In this case, LCM(2, 8) = 8.
• Don't forget the $T_{r+1}$ aspect: The question asks for the number of integral terms, which directly corresponds to the number of valid $r$ values.

---

9. Summary and Key Takeaway

To determine the number of integral terms in a binomial expansion of the form $(\sqrt[p]{X} + \sqrt[q]{Y})^n$:

1. Write out the general term $T_{r+1} = {}^{n}C_r \cdot X^{\frac{n-r}{p}} \cdot Y^{\frac{r}{q}}$.
2. Set up conditions that the exponents $\frac{n-r}{p}$ and $\frac{r}{q}$ must be integers. This means $(n-r)$ must be a multiple of $p$, and $r$ must be a multiple of $q$.
3. Combine these divisibility conditions for $r$.
4. Identify the valid range for $r$ ($0 \le r \le n$).
5. Count the number of $r$ values that satisfy all conditions using the arithmetic progression formula. This count will be the number of integral terms.