Given that `._(1)mu_(2)=4//3`,`._(2)mu_3=3//2.` Find `._(1)_mu_3.`

Explanation

To solve the problem, we need to find the refractive index \(\mu_{1}\mu_{3}\) given the relationships \(\mu_{1}\mu_{2} = \frac{4}{3}\) and \(\mu_{2}\mu_{3} = \frac{3}{2}\).

### Step-by-Step Solution:

1. **Write down the given equations:**
- From the problem, we have:
\[ \mu_{1}\mu_{2} = \frac{4}{3} \quad \text{(Equation 1)} \]
\[ \mu_{2}\mu_{3} = \frac{3}{2} \quad \text{(Equation 2)} \]

2. **Rearrange Equation 1 to express \(\mu_{2}\):**
- From Equation 1, we can express \(\mu_{2}\) in terms of \(\mu_{1}\):
\[ \mu_{2} = \frac{4}{3\mu_{1}} \quad \text{(Equation 3)} \]

3. **Substitute Equation 3 into Equation 2:**
- Now, substitute \(\mu_{2}\) from Equation 3 into Equation 2:
\[ \left(\frac{4}{3\mu_{1}}\right)\mu_{3} = \frac{3}{2} \]

4. **Solve for \(\mu_{3}\):**
- Multiply both sides by \(3\mu_{1}\) to eliminate the fraction:
\[ 4\mu_{3} = \frac{3}{2} \cdot 3\mu_{1} \]
\[ 4\mu_{3} = \frac{9}{2}\mu_{1} \]
- Now, divide both sides by 4:
\[ \mu_{3} = \frac{9}{8}\mu_{1} \]

5. **Find \(\mu_{1}\mu_{3}\):**
- Now, we can find \(\mu_{1}\mu_{3}\):
\[ \mu_{1}\mu_{3} = \mu_{1} \cdot \left(\frac{9}{8}\mu_{1}\right) = \frac{9}{8}\mu_{1}^{2} \]

6. **Final Result:**
- Therefore, the final relationship we have is:
\[ \mu_{1}\mu_{3} = \frac{9}{8} \]

### Summary:
The value of \(\mu_{1}\mu_{3}\) is \(\frac{9}{8}\).