Worksheet for practicing multiplication and division of monomials with exponent rules.

To solve the given problems involving multiplying and dividing monomials, we will simplify each expression step by step. Let's go through each problem:

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Problem 1: \( 2x^3 \cdot 4y^4 \)

Step 1: Multiply the coefficients.
\[ 2 \cdot 4 = 8 \]

Step 2: Combine the variables using the laws of exponents.
- For \( x \): The exponent is \( 3 \) (since there is no other \( x \)).
- For \( y \): The exponent is \( 4 \).

So, the simplified expression is:
\[ 8x^3y^4 \]

Answer: \( 8x^3y^4 \)

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Problem 2: \( \frac{2x^4}{4x^3} \)

Step 1: Simplify the coefficients.
\[ \frac{2}{4} = \frac{1}{2} \]

Step 2: Simplify the variable \( x \) using the quotient rule for exponents (\( \frac{x^m}{x^n} = x^{m-n} \)).
\[ \frac{x^4}{x^3} = x^{4-3} = x^1 = x \]

So, the simplified expression is:
\[ \frac{1}{2}x \]

Answer: \( \frac{x}{2} \)

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Problem 3: \( 4x^2 - 3xy^2 \)

This expression cannot be simplified further because it involves subtraction of unlike terms.

Answer: \( 4x^2 - 3xy^2 \)

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Problem 4: \( (2x^2)(6x^4y^3) \)

Step 1: Multiply the coefficients.
\[ 2 \cdot 6 = 12 \]

Step 2: Combine the variables using the laws of exponents.
- For \( x \): \( x^2 \cdot x^4 = x^{2+4} = x^6 \)
- For \( y \): The exponent is \( 3 \) (since there is no other \( y \)).

So, the simplified expression is:
\[ 12x^6y^3 \]

Answer: \( 12x^6y^3 \)

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Problem 5: \( \frac{9x^{10}y^7}{3x^7y^5} \)

Step 1: Simplify the coefficients.
\[ \frac{9}{3} = 3 \]

Step 2: Simplify the variables using the quotient rule for exponents.
- For \( x \): \( \frac{x^{10}}{x^7} = x^{10-7} = x^3 \)
- For \( y \): \( \frac{y^7}{y^5} = y^{7-5} = y^2 \)

So, the simplified expression is:
\[ 3x^3y^2 \]

Answer: \( 3x^3y^2 \)

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Problem 6: \( \frac{14a^{12}b^8}{7a^8b^2} \)

Step 1: Simplify the coefficients.
\[ \frac{14}{7} = 2 \]

Step 2: Simplify the variables using the quotient rule for exponents.
- For \( a \): \( \frac{a^{12}}{a^8} = a^{12-8} = a^4 \)
- For \( b \): \( \frac{b^8}{b^2} = b^{8-2} = b^6 \)

So, the simplified expression is:
\[ 2a^4b^6 \]

Answer: \( 2a^4b^6 \)

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Problem 7: \( \frac{7x^6y}{x} \)

Step 1: Simplify the coefficients.
The coefficient is \( 7 \).

Step 2: Simplify the variables using the quotient rule for exponents.
- For \( x \): \( \frac{x^6}{x} = x^{6-1} = x^5 \)
- For \( y \): The exponent is \( 1 \) (since there is no other \( y \)).

So, the simplified expression is:
\[ 7x^5y \]

Answer: \( 7x^5y \)

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Problem 8: \( 2x^3y - 6x^2y^{-1} \)

This expression cannot be simplified further because it involves subtraction of unlike terms.

Answer: \( 2x^3y - 6x^2y^{-1} \)

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Problem 9: \( (4x^2y)(2xy^2) \)

Step 1: Multiply the coefficients.
\[ 4 \cdot 2 = 8 \]

Step 2: Combine the variables using the laws of exponents.
- For \( x \): \( x^2 \cdot x = x^{2+1} = x^3 \)
- For \( y \): \( y \cdot y^2 = y^{1+2} = y^3 \)

So, the simplified expression is:
\[ 8x^3y^3 \]

Answer: \( 8x^3y^3 \)

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Problem 10: \( (2x^2y^3)(2x^{-4}) \)

Step 1: Multiply the coefficients.
\[ 2 \cdot 2 = 4 \]

Step 2: Combine the variables using the laws of exponents.
- For \( x \): \( x^2 \cdot x^{-4} = x^{2+(-4)} = x^{-2} \)
- For \( y \): The exponent is \( 3 \) (since there is no other \( y \)).

So, the simplified expression is:
\[ 4x^{-2}y^3 \]

Answer: \( \frac{4y^3}{x^2} \)

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Problem 11: \( \frac{3x^3y^2}{x^4y^3} \)

Step 1: Simplify the coefficients.
\[ \frac{3}{1} = 3 \]

Step 2: Simplify the variables using the quotient rule for exponents.
- For \( x \): \( \frac{x^3}{x^4} = x^{3-4} = x^{-1} \)
- For \( y \): \( \frac{y^2}{y^3} = y^{2-3} = y^{-1} \)

So, the simplified expression is:
\[ 3x^{-1}y^{-1} \]

Answer: \( \frac{3}{xy} \)

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Problem 12: \( \frac{12x^8y^4}{4x^7y^2} \)

Step 1: Simplify the coefficients.
\[ \frac{12}{4} = 3 \]

Step 2: Simplify the variables using the quotient rule for exponents.
- For \( x \): \( \frac{x^8}{x^7} = x^{8-7} = x^1 = x \)
- For \( y \): \( \frac{y^4}{y^2} = y^{4-2} = y^2 \)

So, the simplified expression is:
\[ 3xy^2 \]

Answer: \( 3xy^2 \)

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Problem 13: \( (2x^2y^3)(x^{-1}y) \)

Step 1: Multiply the coefficients.
\[ 2 \cdot 1 = 2 \]

Step 2: Combine the variables using the laws of exponents.
- For \( x \): \( x^2 \cdot x^{-1} = x^{2+(-1)} = x^1 = x \)
- For \( y \): \( y^3 \cdot y = y^{3+1} = y^4 \)

So, the simplified expression is:
\[ 2xy^4 \]

Answer: \( 2xy^4 \)

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Problem 14: \( \frac{(xy)^{-3}}{(xy)^{-2}} \)

Step 1: Use the quotient rule for exponents (\( \frac{a^m}{a^n} = a^{m-n} \)).
\[ \frac{(xy)^{-3}}{(xy)^{-2}} = (xy)^{-3 - (-2)} = (xy)^{-3 + 2} = (xy)^{-1} \]

Step 2: Rewrite with a positive exponent.
\[ (xy)^{-1} = \frac{1}{xy} \]

Answer: \( \frac{1}{xy} \)

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Problem 15: \( (2x^2y^3)(3x^{-1}y)(2xy^{-1}) \)

Step 1: Multiply the coefficients.
\[ 2 \cdot 3 \cdot 2 = 12 \]

Step 2: Combine the variables using the laws of exponents.
- For \( x \): \( x^2 \cdot x^{-1} \cdot x = x^{2 + (-1) + 1} = x^2 \)
- For \( y \): \( y^3 \cdot y \cdot y^{-1} = y^{3 + 1 + (-1)} = y^3 \)

So, the simplified expression is:
\[ 12x^2y^3 \]

Answer: \( 12x^2y^3 \)

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Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & 8x^3y^4 \\
2. & \frac{x}{2} \\
3. & 4x^2 - 3xy^2 \\
4. & 12x^6y^3 \\
5. & 3x^3y^2 \\
6. & 2a^4b^6 \\
7. & 7x^5y \\
8. & 2x^3y - 6x^2y^{-1} \\
9. & 8x^3y^3 \\
10. & \frac{4y^3}{x^2} \\
11. & \frac{3}{xy} \\
12. & 3xy^2 \\
13. & 2xy^4 \\
14. & \frac{1}{xy} \\
15. & 12x^2y^3 \\
\end{array}
}
\]