The total number of irrational terms in the binomial expansion of (7^(1/5) - 3^(1/10))⁶⁰ is : (1) 55

Q: Wait, why is the total number of terms $n+1$ and not $n$?

Great question! In the binomial expansion of $(a+b)^n$, the terms are indexed from $r=0$ to $r=n$. If you count them, $0, 1, 2, ..., n$, that is exactly $n+1$ terms. For $n=60$, we have $60+1=61$ terms.

Q: Does the negative sign in $(7^{1/5} - 3^{1/10})$ affect whether a term is rational?

Not at all! The negative sign just means the term is multiplied by $(-1)^r$. Since $(-1)^r$ is either $1$ or $-1$, it is always a rational number. It does not change the rationality of the term itself.

Q: Why can't we just count the irrational terms directly?

Because the irrational terms are 'messy'—they involve various combinations of roots. It is mathematically much cleaner to identify the 'special' rational terms and subtract them from the total. It is a classic, elegant shortcut in JEE problems.

Question

The total number of irrational terms in the binomial expansion of $(7^{1/5} - 3^{1/10})^{60}$ is :

Options:

(A) $55$

(B) $49$

(C) $48$

(D) $54$

Sigma Percentile · Accepted Answer

### Analyzing the Binomial Landscape

Imagine you are standing before the vast expansion of $(7^{1/5} - 3^{1/10})^{60}$. It is a daunting expression, but as an elite student, you know that we do not brute-force our way through these problems.

We use the elegance of the **Binomial Theorem** to navigate this complexity.

### The Master Strategy: The Art of Subtraction

When faced with a question asking for the number of irrational terms, your first instinct might be to try and count them. Resist that urge!

The irrational terms are like a chaotic sea. Instead, we look for the islands of stability: the rational terms. Our strategy is defined by the following relationship:

$$	ext{Irrational Terms} = 	ext{Total Terms} - 	ext{Rational Terms}$$

We know the total number of terms in the expansion is $n+1$, which is $61$. Now, we simply need to isolate the rational ones.

### Decoding the DNA: The General Term

Every term in this expansion is governed by the general term formula:

$$T_{r+1} = \binom{60}{r} (7^{1/5})^{60-r} (-3^{1/10})^r$$

By applying the laws of exponents, we simplify this to:

$$T_{r+1} = \binom{60}{r} (-1)^r 7^{\frac{60-r}{5}} 3^{\frac{r}{10}}$$

This expression is the DNA of our expansion. For a term to be rational, the exponents of $7$ and $3$ must be integers. If they are not, the term remains trapped in the world of roots and irrationals.

### The Filter of Rationality

We have two conditions that must be satisfied simultaneously:

$1)$ $\frac{60-r}{5} \in \mathbb{Z}$

$2)$ $\frac{r}{10} \in \mathbb{Z}$

Let us look at the second condition first. For $\frac{r}{10}$ to be an integer, $r$ must be a multiple of $10$. Since $r$ ranges from $0$ to $60$, our candidates are $r \in \{0, 10, 20, 30, 40, 50, 60\}$.

Now, we test these against the first condition. If $r$ is a multiple of $10$, then $60-r$ is also a multiple of $10$, which is clearly divisible by $5$. Every single one of our candidates satisfies both conditions.

### The Final Triumph

We have found $7$ values of $r$ that produce rational terms. That means there are exactly $7$ rational terms in this entire expansion.

To find the irrational ones, we simply perform the final subtraction:

$$61 - 7 = 54$$

Through the power of logical deduction, we have navigated the complexity of the binomial expansion to find exactly **54 irrational terms**. Remember, in JEE, it is not about how hard you work, but how smart you look at the problem.

Animated Solution for Mathematics - Binomial Theorem: The total number of irrational terms in the binomial expansion of $(7^{1/5} - 3^{1/10})^{60}$ is :

Select Answer:

Visualized Solution

The Objective

Strategy Overview

The General Term $T_{r + 1}$

Substitution

Simplifying Exponents

The Rationality Condition

Analyzing the Conditions

Solving for $r$

Counting Rational Terms

Total Number of Terms

The Final Subtraction

The Sigma Insight: General Term and Middle Term

Analyzing the Binomial Landscape

The Master Strategy

Decoding the DNA

The Filter of Rationality

The Final Triumph

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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 total number of irrational terms in the binomial expansion of (71/5−31/10)60 is :