From the image you provided, it looks like you're simplifying the expression

$\left(\frac{12x^5 y^4}{4x^8 y^{-1}}\right)^2$

You entered:

$3x^{-8}y^5$

However, the feedback suggests that the powers must be positive, meaning your result contains a negative exponent that needs adjusting. Let's go step-by-step through the simplification process:

Step 1: Simplify the fraction inside the parentheses.

Start by simplifying the expression inside the parentheses.

$\frac{12x^5 y^4}{4x^8 y^{-1}}$

1. Simplify the coefficients: $\frac{12}{4} = 3$

2. Simplify the $x$-terms by subtracting exponents: $\frac{x^5}{x^8} = x^{5-8} = x^{-3}$

3. Simplify the $y$-terms by subtracting exponents: $\frac{y^4}{y^{-1}} = y^{4-(-1)} = y^{4+1} = y^5$

So, after simplifying, we get:

$3x^{-3}y^5$

Step 2: Square the result.

Now, square the entire simplified expression:

$\left(3x^{-3}y^5\right)^2$

1. Square the coefficient: $3^2 = 9$

2. Square the $x$-term: $(x^{-3})^2 = x^{-6}$

3. Square the $y$-term: $(y^5)^2 = y^{10}$

So, the final simplified expression is:

$9x^{-6}y^{10}$

Step 3: Ensure positive exponents.

To make all the exponents positive, write $x^{-6}$ as $\frac{1}{x^6}$:

$\frac{9y^{10}}{x^6}$

This is the fully simplified expression with all positive exponents.

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you simplify expressions with negative exponents?
2. What is the general rule for dividing powers with the same base?
3. How do you handle coefficients when simplifying expressions?
4. Why is it important to express the final result with positive exponents?
5. Can you apply these rules to more complex rational expressions?

Tip: Always simplify the expression inside parentheses before applying any exponent outside!