https://file.aiquickdraw.com/mathbot/file/17258136221171vdo0bde.jpg

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.

Simplify the expression: (x^(-4) / y^3)^2 ÷ (y/x)^6

Learn how to simplify complex exponential expressions involving fractional and negative exponents, such as (x^(-4) / y^3)^2 ÷ (y/x)^6. This problem covers key algebraic concepts, including exponentiation rules and operations. Suitable for students in Grades 9-10.

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