The term independent of x in `(2x^(1//2)-3x^(-1//3))^20`

Options

A. `""^20C_(7) 2^(8) 3^12`
B. `""^20C_(8) 2^(7) 3^13`
C. `""^(20)C_(8) C^(8) 3^12`
D. `""^20C_(7)2^(7) 3^13`

Answer

To find the term independent of \( x \) in the expression \( (2x^{1/2} - 3x^{-1/3})^{20} \), we can follow these steps:

### Step 1: Identify the General Term
The general term in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
In our case, \( a = 2x^{1/2} \), \( b = -3x^{-1/3} \), and \( n = 20 \). Therefore, the general term \( T_{r+1} \) can be expressed as:
\[ T_{r+1} = \binom{20}{r} (2x^{1/2})^{20-r} (-3x^{-1/3})^r \]

### Step 2: Simplify the General Term
Now, we simplify \( T_{r+1} \):
\[ T_{r+1} = \binom{20}{r} \cdot 2^{20-r} \cdot (-3)^r \cdot x^{(20-r)/2} \cdot x^{-r/3} \]
Combining the powers of \( x \):
\[ T_{r+1} = \binom{20}{r} \cdot 2^{20-r} \cdot (-3)^r \cdot x^{(20-r)/2 - r/3} \]

### Step 3: Set the Power of \( x \) to Zero
For the term to be independent of \( x \), the exponent of \( x \) must be zero:
\[ \frac{20 - r}{2} - \frac{r}{3} = 0 \]
To eliminate the fractions, we can multiply through by 6 (the least common multiple of 2 and 3):
\[ 6 \left( \frac{20 - r}{2} \right) - 6 \left( \frac{r}{3} \right) = 0 \]
This simplifies to:
\[ 3(20 - r) - 2r = 0 \]
Expanding this gives:
\[ 60 - 3r - 2r = 0 \]
Combining like terms:
\[ 60 - 5r = 0 \]
Solving for \( r \):
\[ 5r = 60 \implies r = 12 \]

### Step 4: Substitute \( r \) Back into the General Term
Now that we have \( r = 12 \), we can find the term \( T_{13} \) (since \( r+1 = 13 \)):
\[ T_{13} = \binom{20}{12} (2x^{1/2})^{20-12} (-3x^{-1/3})^{12} \]
Calculating this gives:
\[ T_{13} = \binom{20}{12} (2x^{1/2})^8 (-3x^{-1/3})^{12} \]
\[ = \binom{20}{12} \cdot 2^8 \cdot (-3)^{12} \cdot x^{(8/2) + (-12/3)} \]
\[ = \binom{20}{12} \cdot 2^8 \cdot (-3)^{12} \cdot x^{4 - 4} = \binom{20}{12} \cdot 2^8 \cdot (-3)^{12} \]

### Step 5: Calculate the Coefficient
The coefficient of the term independent of \( x \) is:
\[ \binom{20}{12} \cdot 2^8 \cdot (-3)^{12} \]
Calculating \( \binom{20}{12} = \binom{20}{8} \) (since \( \binom{n}{r} = \binom{n}{n-r} \)):
\[ \binom{20}{8} = 125970 \]
Calculating \( 2^8 = 256 \) and \( (-3)^{12} = 531441 \):
\[ \text{Coefficient} = 125970 \cdot 256 \cdot 531441 \]

### Final Result
The term independent of \( x \) is:
\[ 125970 \cdot 256 \cdot 531441 \]