The number of integral terms in the expansion of (√3 + ⁸√5)²⁵⁶ is (a) 35 (b) 32 (c) 33 (d) 34

Q: Wait, why do we include $r=0$? Isn't the first term just $\binom{256}{0} a^{256}$?

Excellent observation! Many students skip $r=0$ because they think it's 'not a term'. But in the expansion of $(a+b)^n$, the terms are indexed from $r=0$ to $r=n$. If $r=0$ satisfies the condition, it is absolutely an integral term. Always check the boundaries!

Q: Why can't we just use energy conservation or some other physics trick?

Haha, I love the enthusiasm! But this is pure algebra. We are looking for the 'integral' nature of terms. The only 'energy' here is the power of the binomial theorem. Stick to the prime factorization logic; it's the most robust tool in your arsenal.

Q: What if the bases were not prime numbers, like $6$ and $10$?

That is a fantastic question! If the bases were composite, you would first decompose them into their prime factors (e.g., $6 = 2 \times 3$). Then, you would ensure the exponent of *each* prime factor is a non-negative integer. It adds a layer of complexity, but the core principle remains identical.

Question

The number of integral terms in the expansion of $(\sqrt{3} + \sqrt[8]{5})^{256}$ is

Options:

(A) 35

(B) 32

(C) 33

(D) 34

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before the massive expansion of $(\sqrt{3} + \sqrt[8]{5})^{256}$. It is a daunting expression, isn't it? If you were to expand this, you would have $257$ individual terms.

But we don't need to write them all out. We are on a treasure hunt to find the 'integral' terms—the ones that are perfect integers.

### The General Term: Our Master Key

In the world of binomial expansions, we have a secret weapon: the general term formula. For any expansion $(a+b)^n$, the $(r+1)$-th term is given by:

$$T_{r+1} = \binom{n}{r} a^{n-r} b^r$$

This formula is our scout; it allows us to peek at any term without doing the heavy lifting of full expansion. Here, our $n$ is $256$, our $a$ is $3^{1/2}$, and our $b$ is $5^{1/8}$.

Substituting these into our formula, we get:

$$T_{r+1} = \binom{256}{r} (3^{1/2})^{256-r} (5^{1/8})^r$$

### Simplifying the Exponents

Now, let's clean up the exponents using the law of indices $(x^m)^n = x^{mn}$. The term becomes:

$$T_{r+1} = \binom{256}{r} 3^{\frac{256-r}{2}} 5^{\frac{r}{8}}$$

This is where the magic happens. For $T_{r+1}$ to be an integer, the exponents of our prime bases, $3$ and $5$, must be non-negative integers.

The binomial coefficient $\binom{256}{r}$ is always an integer, so we don't need to worry about it. We focus entirely on the exponents: $\frac{256-r}{2}$ and $\frac{r}{8}$.

### The Constraints: Solving the Puzzle

For the exponent of $3$ to be an integer, $\frac{256-r}{2} = 128 - \frac{r}{2}$ must be an integer. Since $128$ is an integer, $\frac{r}{2}$ must be an integer, meaning $r$ must be a multiple of $2$.

For the exponent of $5$ to be an integer, $\frac{r}{8}$ must be an integer, meaning $r$ must be a multiple of $8$. To satisfy both conditions, $r$ must be a multiple of the Least Common Multiple of $2$ and $8$, which is $8$.

Thus, $r$ must be a multiple of $8$.

### The Final Count: The Rhythm of the Sequence

We know that in a binomial expansion, $r$ ranges from $0$ to $n$. So, $0 \le r \le 256$. We are looking for multiples of $8$ in this range: $0, 8, 16, \dots, 256$.

This is an Arithmetic Progression where the first term $a=0$, the common difference $d=8$, and the last term $l=256$. The number of terms is calculated as:

$$n = \frac{l-a}{d} + 1 = \frac{256-0}{8} + 1 = 32 + 1 = 33$$

And there we have it! Out of the $257$ terms, exactly **$33$** are integers. It is a beautiful, elegant result that rewards our logical persistence.

Animated Solution for Mathematics - Binomial Theorem: The number of integral terms in the expansion of $(3 + 85)^{256}$ is

Select Answer:

Visualized Solution

The Binomial Expression

General Term Formula $T_{r + 1}$

Substitute the Values

Simplify the Exponents

Condition for Integral Terms

Analyze the First Exponent

Analyze the Second Exponent

Combine the Conditions

Define the Range of $r$

Identify Valid Values of $r$

Recognize the Arithmetic Progression

Calculate the Number of Terms

Final Computation

Final Conclusion

The Sigma Insight: Binomial Expansion for Positive Integral Index

Analyzing the Setup

The General Term

Simplifying the Exponents

The Constraints

The Final Count

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Animated Solution for Mathematics - Binomial Theorem: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of integral terms in the expansion of (3​+85​)256 is