If $\frac{x}{y}= (\frac{-1}{3})^{-3}$ $\div(\frac{2}{3})^{-4}$, then what is the value of ($\frac{x}{y}+\frac{y}{x}$)$^{-1}$?

Question

If  $\frac{x}{y}= (\frac{-1}{3})^{-3}$ $\div(\frac{2}{3})^{-4}$, then what is the value of ($\frac{x}{y}+\frac{y}{x}$)$^{-1}$?

Accepted Answer

To solve the given problem, we need to evaluate the expression $\frac{x}{y} = \left(\frac{-1}{3}\right)^{-3} \div \left(\frac{2}{3}\right)^{-4}$ and then find the value of $\left(\frac{x}{y} + \frac{y}{x}\right)^{-1}$.

### Step-by-Step Solution:

1. **Evaluate $\frac{x}{y}$:**

- First, calculate $\left(\frac{-1}{3}\right)^{-3}$:
     $$
     \left(\frac{-1}{3}\right)^{-3} = \left(-3\right)^3 = -27
     $$

- Next, calculate $\left(\frac{2}{3}\right)^{-4}$:
     $$
     \left(\frac{2}{3}\right)^{-4} = \left(\frac{3}{2}\right)^4 = \frac{81}{16}
     $$

- Now, divide the two results:
     $$
     \frac{x}{y} = \frac{-27}{1} \div \frac{81}{16} = -27 \times \frac{16}{81} = -\frac{432}{81} = -\frac{16}{3}
     $$

2. **Calculate $\frac{x}{y} + \frac{y}{x}$:**

- We have $\frac{x}{y} = -\frac{16}{3}$, so $\frac{y}{x} = -\frac{3}{16}$.

- Add these two fractions:
     $$
     \frac{x}{y} + \frac{y}{x} = -\frac{16}{3} + \left(-\frac{3}{16}\right)
     $$

- Find a common denominator and add:
     $$
     = -\frac{16 \times 16}{48} - \frac{3 \times 3}{48} = -\frac{256}{48} - \frac{9}{48} = -\frac{265}{48}
     $$

3. **Find the inverse:**

- The inverse of $-\frac{265}{48}$ is:
     $$
     \left(\frac{x}{y} + \frac{y}{x}\right)^{-1} = -\frac{48}{265}
     $$

### Conclusion:

The value of $\left(\frac{x}{y} + \frac{y}{x}\right)^{-1}$ is $-\frac{48}{265}$, which corresponds to Option 4.

### Explanation of Incorrect Options:

- **Option 1: -$\frac{3}{16}$**

This option does not match the calculated inverse value of $-\frac{48}{265}$.

- **Option 2: $\frac{19}{48}$**

This option is a positive fraction and does not match the negative inverse value calculated.

- **Option 3: $\frac{38}{73}$**

This option is also a positive fraction and does not match the negative inverse value calculated.

Thus, Option 4 is the only correct answer, as it matches the calculated result of $-\frac{48}{265}$.