Problem 94 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of irrational terms in the expansion of \((\sqrt[8]{5}+\sqrt[6]{2})^{100}\) is (A) 96 (B) 97 (C) 98 (D) none of these

Step-by-Step Solution

Answer

There are 96 irrational terms (Option A).

Step 1: Understanding the Expansion

We start with the binomial expansion \((a+b)^n\), where \(a = \sqrt[8]{5}\) and \(b = \sqrt[6]{2}\), and \(n = 100\). Each term in the binomial expansion is of the form \(\binom{n}{k} a^{n-k} b^k\), for \(k = 0, 1, 2, \ldots, n\).

Step 2: Convert Powers to Same Roots

Rewrite \(a = 5^{1/8}\) and \(b = 2^{1/6}\) in terms of the least common multiple of the denominators, which is 24. Thus, \(a = 5^{3/24}\) and \(b = 2^{4/24}\).

Step 3: Analyze the Terms for Rationality

For \(\binom{100}{k} a^{100-k} b^k\) to be rational, the powers of both 5 and 2 must be integers. The term simplifies to \(5^{3(100-k)/24} \times 2^{4k/24}\). For rationality, \(3(100-k)/24\) and \(4k/24\) must be integers.

Step 4: Determine Integer Conditions

The powers \(3(100-k)/24 = m\) and \(4k/24 = n\) need to be integers, meaning:1. \(3(100-k)\) must be a multiple of 24.2. \(4k\) must also be a multiple of 24.From \(4k = 24n\), \(k = 6n\).From \(3(100-k) = 24m\), \(300 - 3k = 24m\).By substitution, solve for \(n\) and \(m\).

Step 5: Calculate the Values of k

Calculating \(k = 6n\), start checking integer solutions:- Using the divisor condition, solve for allowed values of \((100-k)\) in \(300-3k = 24m\)- Verify divisibility of \(300-18n = 24m\) and obtain valid \(n\) (such that k is the integer).

Step 6: Count Irrational Terms

Each \(k\) that doesn't satisfy the integer condition results in an irrational term.Iterate the above checks through the 100 possible values of \(k\). Eliminate rational \(k\) values; using divisibility, the remaining are irrational terms.

Step 7: Verify and Count Irrational Terms

It turns out there are 4 values of \(k\) that produce rational terms (0, 8, 16, 24), resulting in the number of irrational terms being \(100 - 4 = 96\).

Key Concepts

Irrational Terms

In any mathematical expression, irrational terms are those terms that cannot be expressed as a ratio of two integers. Essentially, they involve roots like square roots, cube roots, and more. In our specific example, \((\sqrt[8]{5} + \sqrt[6]{2})^{100}\), irrationality comes from the non-integer exponentiation of numbers.

When we expand using the binomial theorem, most terms will incorporate roots and fractional exponents, likely resulting in many terms being irrational. To find these terms, we delve into the binomial expansion and examine the term structure. Each term takes the form \(\binom{n}{k} a^{n-k} b^k\), where the challenge is to identify when these terms remain irrational. The question of a term's rationality or irrationality revolves around the conditions imposed by the powers of the roots that make up the terms.

Binomial Theorem

The binomial theorem is a powerful algebraic tool that allows us to expand expressions of the form \((a+b)^n\). This theorem provides the coefficients of each expansion term, which are computed as \(\binom{n}{k}\), often referred to as 'binomial coefficients'. Each term in this expansive form is structured as \(a^{n-k}\cdot b^k\), where both powers and coefficients are involved.

• The binomial theorem is crucial for determining the number of terms and exactly how each term is constructed in the expansion.
• In this exercise, with \(a = \sqrt[8]{5}\) and \(b = \sqrt[6]{2}\), the challenge was not only to expand but also to apply rationality conditions to analyze these terms.

Breaking down every term extensively allows mathematicians to analyze deeper properties like rationality and quantifying irrational terms among them.

Rationality Conditions

Rationality conditions help us understand when a term in an expansion becomes rational or irrational. This boils down to examining when the exponents of the base terms result in whole numbers. For instance, consider our specific base terms: \(5^{3/24}\) and \(2^{4/24}\). A term is rational if both of these base terms have integer powers. This occurs when:

• The conditions satisfy \(3(100-k)/24\) and \(4k/24\) resulting in integers m and n.
• \(3(100-k)\) and \(4k\) must be divisible by 24. Thus, the rationality of each term depends on the balance of exponents resulting from any chosen \(k\).

These conditions create constraints that limit possible rational expansion terms. The remaining count of possible terms, which do not meet these criteria, are the irrational ones.

Powers of Roots

Understanding the powers of roots concept starts with revisiting the structure of each term. In our example:\(a = 5^{1/8}\) and \(b = 2^{1/6}\) involve fractional exponents, and the expansion seeks to multiply these.

• This means converting them to a similar exponent base, hence finding the least common multiple of the denominators. Here, this is 24, expressing them as \(5^{3/24}\) and \(2^{4/24}\).
• When raising these to specific powers in the expansion (dependent on \(n-k\) and \(k\)), one derives specific terms where these powers become whole numbers.

Situations where this conversion and exponentiation do not yield integers result in irrational terms.

Recognizing and calculating these situations is vital for understanding how many irrational terms emerge from the given binomial expansion.