Problem 123 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of integral terms in the expansion of \((\sqrt{3}+\sqrt[8]{5})^{256}\) is (A) 32 (B) 33 (C) 34 (D) 35

Step-by-Step Solution

Answer

There are 33 integral terms in the expansion.

Step 1: Understanding the Problem

We need to find how many integral terms there are in the binomial expansion of \((\sqrt{3}+\sqrt[8]{5})^{256}\). An integral term occurs when the powers of both components are integers.

Step 2: Setting Up the General Term

In the expansion of \((\sqrt{3}+\sqrt[8]{5})^{256}\), the general term can be expressed as \(T_{k+1} = \binom{256}{k} (\sqrt{3})^{256-k} (\sqrt[8]{5})^k\). We need integer terms, so we require both powers to be integers.

Step 3: Calculating the Conditions for Integer Powers

The term \((\sqrt{3})^{256-k}\) becomes an integer if \(256-k\) is even, meaning \(256-k\equiv 0 \pmod{2}\), or \(k\equiv 0 \pmod{2}\). Similarly, \((\sqrt[8]{5})^k\) is an integer if \(k\equiv 0 \pmod{8}\).

Step 4: Common Condition for Integer Power

For both conditions to be satisfied, \(k\) should be a multiple of both 2 and 8. Thus, \(k\) must be a multiple of the least common multiple of 2 and 8, which is 8, i.e., \(k\equiv 0 \pmod{8}\).

Step 5: Finding Eligible Values for k

The values of \(k\) that are multiples of 8 range from 0 to 256. These are \(k = 0, 8, 16, ..., 256\), which is an arithmetic sequence where \(a_1 = 0\) and \(d = 8\).

Step 6: Counting the Terms

The number of terms is given by the number of terms in the arithmetic sequence \(k = 0, 8, 16, ..., 256\). The last term \(k_n = 256\) and we use the formula for the number of terms in an arithmetic sequence: \(n = \frac{k_n - a_1}{d} + 1\).

Step 7: Calculating the Number of Terms

Substitute \(k_n = 256\), \(a_1 = 0\), and \(d = 8\) into the formula: \[ n = \frac{256 - 0}{8} + 1 = 32 + 1 = 33. \] So, there are 33 integral terms.

Key Concepts

Understanding Binomial Expansion

Binomial expansion is a method that allows the expansion of expressions raised to a power. The concept is based on the binomial theorem, which states that for any integer power \( n \), \[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k.\] The coefficients \( \binom{n}{k} \) are known as binomial coefficients, calculated using combinations.

• The term under expansion involves a sum of two variables or numbers.
• Each term in the expanded form consists of a product of a coefficient and powers of the two variables.

In the given problem, we use this expansion to determine specific integer-valued terms, which means paying attention to the conditions that transform the roots into whole numbers.

Exploring Integer Powers

Integer powers are required when the exponents of each element result in an integer value. It involves determining under what cycle or condition a root raised to a power becomes a whole number. Looking at \( (\sqrt{3})^{256-k} \), an integer term occurs when the power \( 256-k \) is even, since the square root of 3 raised to an even power results in an integer.

• If the power is odd, the term remains a root, not an integer.

Similarly for \( (\sqrt[8]{5})^k \), which becomes an integer only if \( k \) is a multiple of 8. This is due to the 8th root of 5 needing to be raised to a power that nullifies the fractional part. Both conditions check the exponent's parity.

Counting Terms in an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms. To find eligible terms for binomial expansion that are integers, we formed an arithmetic sequence based on the conditions \( k \equiv 0 \pmod{8} \).

• The sequence starts at 0 (\( a_1 = 0 \)).
• It increases by 8 each step (difference \( d = 8 \)).
• Continues up to 256.

To find the number of such terms, we use the formula: \[ n = \frac{k_n - a_1}{d} + 1, \]where \( k_n \) is the last term. Once applied, we find there are 33 integral terms using this sequence.