The expression in the image is:

$\left(\frac{x^{-4}}{y^3}\right)^2 \div \left(\frac{y}{x}\right)^6$

Let's simplify it step by step.

Step 1: Simplify the first part

The expression inside the first parentheses is $\frac{x^{-4}}{y^3}$. When you square this expression, you raise both the numerator and the denominator to the power of 2:

$\left(\frac{x^{-4}}{y^3}\right)^2 = \frac{(x^{-4})^2}{(y^3)^2} = \frac{x^{-8}}{y^6}$

Step 2: Simplify the second part

Now, simplify the second part $\frac{y}{x}$ raised to the 6th power:

$\left(\frac{y}{x}\right)^6 = \frac{y^6}{x^6}$

Step 3: Division of the two fractions

Now we divide the first simplified expression by the second one:

$\frac{\frac{x^{-8}}{y^6}}{\frac{y^6}{x^6}} = \frac{x^{-8}}{y^6} \times \frac{x^6}{y^6}$

Step 4: Multiply the fractions

When multiplying fractions, multiply the numerators and the denominators:

$\frac{x^{-8} \cdot x^6}{y^6 \cdot y^6} = \frac{x^{-2}}{y^{12}}$

Final Answer

So, the simplified expression is:

$\frac{x^{-2}}{y^{12}} = \frac{1}{x^2 y^{12}}$

Would you like any more details or have any further questions?

Here are 5 related questions that you might find useful:

1. How would the result change if the original expression had a different exponent?
2. Can you explain the process of multiplying exponents when the bases are the same?
3. What would the final expression look like if $x$ and $y$ were negative?
4. How do exponent rules change when dealing with fractional exponents?
5. How does this process change if there are additional variables involved?

Tip: Always remember to handle negative exponents carefully; they often mean you should take the reciprocal of the base raised to the positive exponent.