A nucleus disintegrates into two smaller parts,which have their velocities in the ratio `3:2`.The ratio of their nudear sizes will be `((x)/(3))^((1)/(3))`.The value of `x` is :

Answer

To solve the problem, we will use the principles of conservation of momentum and the relationship between the sizes of the nuclei and their velocities.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a nucleus that disintegrates into two smaller parts. Let the masses of the two parts be \( m_1 \) and \( m_2 \), and their velocities be \( v_1 \) and \( v_2 \) respectively. The problem states that the ratio of their velocities is \( \frac{v_1}{v_2} = \frac{3}{2} \).

2. **Applying Conservation of Momentum**:
According to the conservation of momentum:
\[ m_1 v_1 = m_2 v_2 \]

3. **Expressing Masses in Terms of Densities and Sizes**:
The mass of a nucleus can be expressed in terms of its density (\( \rho \)) and volume (\( V \)). The volume of a nucleus is proportional to the cube of its radius (\( r \)):
\[ m_1 = \rho V_1 = \rho \left( \frac{4}{3} \pi r_1^3 \right) \quad \text{and} \quad m_2 = \rho V_2 = \rho \left( \frac{4}{3} \pi r_2^3 \right) \]
Thus, we can write:
\[ \frac{m_1}{m_2} = \frac{r_1^3}{r_2^3} \]

4. **Substituting into the Momentum Equation**:
From the momentum conservation equation:
\[ m_1 v_1 = m_2 v_2 \implies \frac{m_1}{m_2} = \frac{v_2}{v_1} \]
Substituting the expression for masses:
\[ \frac{r_1^3}{r_2^3} = \frac{v_2}{v_1} \]

5. **Using the Given Velocity Ratio**:
Since \( \frac{v_1}{v_2} = \frac{3}{2} \), we have:
\[ \frac{v_2}{v_1} = \frac{2}{3} \]
Therefore:
\[ \frac{r_1^3}{r_2^3} = \frac{2}{3} \]

6. **Finding the Ratio of Sizes**:
Taking the cube root of both sides gives:
\[ \frac{r_1}{r_2} = \left( \frac{2}{3} \right)^{\frac{1}{3}} \]

7. **Relating to the Given Ratio**:
The problem states that the ratio of their nuclear sizes is given by:
\[ \frac{x}{3}^{\frac{1}{3}} \]
Setting the two expressions equal:
\[ \left( \frac{2}{3} \right)^{\frac{1}{3}} = \frac{x}{3}^{\frac{1}{3}} \]

8. **Cubing Both Sides**:
Cubing both sides results in:
\[ \frac{2}{3} = \frac{x}{3} \]

9. **Solving for \( x \)**:
Multiplying both sides by 3 gives:
\[ x = 2 \]

### Final Answer:
The value of \( x \) is \( 2 \).