Which of the following expansion will have term containing `x^(3)`

Options

A. `(x^(-1//5) +2x^(3//5))^(25)`
B. `(x^(3//5) +2x^(-1//5))^(24)`
C. `(x^(3//5) -2x^(-1//5))^(23)`
D. `(x^(3//5) +2x^(-1//5))^(22)`

Answer

To determine which of the given expansions will have a term containing \( x^3 \), we will analyze each expansion step by step. Let's start with the first option provided in the video transcript.

### Step-by-Step Solution

**Step 1: Identify the Expansion**
The first expansion is given as:
\[ \left( x^{-\frac{1}{5}} + 2x^{\frac{3}{5}} \right)^{25} \]
We need to find the term that contains \( x^3 \).

**Step 2: Apply the Binomial Theorem**
According to the Binomial Theorem, the expansion of \( (A + B)^n \) is given by:
\[ \sum_{r=0}^{n} \binom{n}{r} A^{n-r} B^r \]
In our case, \( A = x^{-\frac{1}{5}} \), \( B = 2x^{\frac{3}{5}} \), and \( n = 25 \).

**Step 3: Write the General Term**
The general term \( T_r \) in the expansion can be expressed as:
\[ T_r = \binom{25}{r} (x^{-\frac{1}{5}})^{25-r} (2x^{\frac{3}{5}})^r \]
This simplifies to:
\[ T_r = \binom{25}{r} 2^r x^{-\frac{25-r}{5}} x^{\frac{3r}{5}} = \binom{25}{r} 2^r x^{-\frac{25}{5} + \frac{3r}{5}} = \binom{25}{r} 2^r x^{-\frac{5}{1} + \frac{3r}{5}} \]

**Step 4: Simplify the Power of \( x \)**
Now, we need to simplify the exponent of \( x \):
\[ -\frac{25}{5} + \frac{3r}{5} = -5 + \frac{3r}{5} \]
We want this exponent to equal \( 3 \):
\[ -5 + \frac{3r}{5} = 3 \]

**Step 5: Solve for \( r \)**
To find \( r \), we solve the equation:
\[ \frac{3r}{5} = 3 + 5 \]
\[ \frac{3r}{5} = 8 \]
Multiplying both sides by \( 5 \):
\[ 3r = 40 \]
Dividing by \( 3 \):
\[ r = \frac{40}{3} \]
Since \( r \) must be a non-negative integer, we check if \( r \) can be an integer. As \( \frac{40}{3} \) is not an integer, we will check if there are any integer values for \( r \) that satisfy the condition.

**Step 6: Check Integer Values**
We can check integer values for \( r \) from \( 0 \) to \( 25 \) to see if any yield \( x^3 \). However, since \( r = \frac{40}{3} \) is not an integer, we conclude that there is no integer \( r \) that satisfies this condition.

### Conclusion
Thus, the first expansion does not contain a term with \( x^3 \).