Let's go through each of the parts in the image and simplify the expressions, ensuring the answers contain only positive exponents.

a) $4x^4 y^3 - xy^4$

Step 1: Factor out the greatest common factor (GCF) from the two terms.

• The GCF of $4x^4 y^3$ and $xy^4$ is $xy^3$.

Thus, factoring out $xy^3$:

$xy^3(4x^3 - y)$

This is the simplified form.

b) $\frac{4xy^2}{-2x^3 y^{-4}}$

Step 1: Simplify the coefficients:

$\frac{4}{-2} = -2$

Step 2: Apply the quotient rule for the exponents. Subtract the exponents of like terms.

For $x$, we have $x^1$ in the numerator and $x^3$ in the denominator:

$x^{1 - 3} = x^{-2}$

For $y$, we have $y^2$ in the numerator and $y^{-4}$ in the denominator:

$y^{2 - (-4)} = y^{6}$

Thus, the expression simplifies to:

$-2x^{-2}y^6$

Convert the negative exponent into a positive exponent:

$\frac{-2y^6}{x^2}$

c) $\left( x^2 y^{-1} \right)^3 \cdot 2x^{-4} y^{-4}$

Step 1: Simplify $\left( x^2 y^{-1} \right)^3$ by applying the power rule $(a^m)^n = a^{mn}$:

$x^{2 \cdot 3} y^{-1 \cdot 3} = x^6 y^{-3}$

Step 2: Now, combine with $2x^{-4} y^{-4}$:

$x^6 y^{-3} \cdot 2x^{-4} y^{-4} = 2x^{6 - 4} y^{-3 - 4}$

This simplifies to:

$2x^2 y^{-7}$

Convert the negative exponent into a positive exponent:

$\frac{2x^2}{y^7}$

d) $\left( x^{-1} y^2 \right)^{-1} \cdot x^{-2} y^2$

Step 1: Simplify $\left( x^{-1} y^2 \right)^{-1}$ by applying the power rule:

$x^{-1 \cdot (-1)} y^{2 \cdot (-1)} = x^1 y^{-2}$

Step 2: Now, combine with $x^{-2} y^2$:

$x^1 y^{-2} \cdot x^{-2} y^2 = x^{1 - 2} y^{-2 + 2}$

This simplifies to:

$x^{-1} y^0 = x^{-1}$

Convert the negative exponent into a positive exponent:

$\frac{1}{x}$

Summary of Simplified Expressions:

a) $xy^3(4x^3 - y)$
b) $\frac{-2y^6}{x^2}$
c) $\frac{2x^2}{y^7}$
d) $\frac{1}{x}$

Would you like further details or have any questions about any of the steps? Here are some related problems you might find useful:

1. Simplify $\frac{8x^3 y^{-5}}{-4x^{-2} y^2}$.
2. Factor and simplify $9x^5 y^4 - 3x^3 y^2$.
3. Simplify $\left( 3x^2 y^{-1} \right)^2 \cdot x^{-4} y^3$.
4. Simplify $\frac{5x^{-2} y^4}{-15x^3 y^{-1}}$.
5. Combine like terms and simplify: $6x^3 y^{-2} - 4x^3 y^{-2} + x^3 y^{-2}$.

Tip: When simplifying expressions with exponents, remember that negative exponents indicate reciprocals, and positive exponents represent direct multiplication.