https://file.aiquickdraw.com/mathbot/file/1727331013217cv8l76e5.jpg

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.

Simplify the following expressions, ensuring your answer contains only positive exponents: a) 4x^4 y^3 - xy^4 b) 4xy^2 / -2x^3 y^-4 c) (x^2 y^-1)^3 · 2x^-4 y^-4 d) (x^-1 y^2)^-1 · x^-2 y^2

Learn how to simplify algebraic expressions involving exponents with a step-by-step solution. This problem covers key algebraic concepts such as the power rule, quotient rule, and negative exponents. Suitable for students in Grades 9-11.

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