In the expansion of `(2-3x^3)^(20)`, if the ratio of `10^(th)` term to `11^(th)` term is `45/22` then `x=`

Options

A. `-(2)/(3)`
B. `(-3)/(2)`
C. `root(-3)(2/3)`
D. `root(-3)(3/2)`

Answer

To solve the problem, we need to find the value of \( x \) in the expansion of \( (2 - 3x^3)^{20} \) given that the ratio of the 10th term to the 11th term is \( \frac{45}{22} \).

### Step-by-Step Solution:

1. **Identify the Terms**:
The general term \( T_r \) in the expansion of \( (a + b)^n \) is given by:
\[ T_r = \binom{n}{r-1} a^{r-1} b^{n - (r-1)} \]
Here, \( a = 2 \), \( b = -3x^3 \), and \( n = 20 \).

2. **Find the 10th Term (\( T_{10} \))**:
For the 10th term (\( r = 10 \)):
\[ T_{10} = \binom{20}{9} (2)^{9} (-3x^3)^{11} \]
Simplifying this:
\[ T_{10} = \binom{20}{9} \cdot 2^9 \cdot (-3)^{11} \cdot (x^3)^{11} \]

3. **Find the 11th Term (\( T_{11} \))**:
For the 11th term (\( r = 11 \)):
\[ T_{11} = \binom{20}{10} (2)^{10} (-3x^3)^{10} \]
Simplifying this:
\[ T_{11} = \binom{20}{10} \cdot 2^{10} \cdot (-3)^{10} \cdot (x^3)^{10} \]

4. **Set Up the Ratio**:
According to the problem, the ratio of the 10th term to the 11th term is given as:
\[ \frac{T_{10}}{T_{11}} = \frac{45}{22} \]

5. **Substitute the Terms**:
Substitute \( T_{10} \) and \( T_{11} \) into the ratio:
\[ \frac{\binom{20}{9} \cdot 2^9 \cdot (-3)^{11} \cdot (x^3)^{11}}{\binom{20}{10} \cdot 2^{10} \cdot (-3)^{10} \cdot (x^3)^{10}} = \frac{45}{22} \]

6. **Simplify the Ratio**:
Simplifying the left-hand side:
\[ \frac{\binom{20}{9}}{\binom{20}{10}} \cdot \frac{2^9}{2^{10}} \cdot \frac{(-3)^{11}}{(-3)^{10}} \cdot \frac{(x^3)^{11}}{(x^3)^{10}} = \frac{45}{22} \]
This simplifies to:
\[ \frac{\binom{20}{9}}{\binom{20}{10}} \cdot \frac{1}{2} \cdot (-3) \cdot x^3 = \frac{45}{22} \]

7. **Calculate the Binomial Coefficient Ratio**:
We know that:
\[ \frac{\binom{20}{9}}{\binom{20}{10}} = \frac{10}{20 - 10} = \frac{10}{10} = 1 \]
Thus, the equation simplifies to:
\[ \frac{-3}{2} x^3 = \frac{45}{22} \]

8. **Solve for \( x^3 \)**:
Rearranging gives:
\[ -3x^3 = \frac{45}{22} \cdot 2 \]
\[ -3x^3 = \frac{90}{22} \]
\[ x^3 = -\frac{90}{66} = -\frac{15}{11} \]

9. **Find \( x \)**:
Taking the cube root:
\[ x = -\sqrt[3]{\frac{15}{11}} \]

### Final Answer:
\[ x = -\sqrt[3]{\frac{15}{11}} \]