Worksheet for practicing multiplication of monomials with examples and exercises.

Problem Overview:

The task involves multiplying monomials. A monomial is a single term consisting of coefficients, variables, and exponents. To multiply monomials, we follow these steps:
1. Multiply the coefficients (the numerical parts).
2. Add the exponents of the same variables.

Let's solve each problem step by step.

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Example Solution:

The example provided is:
$$
(4x^3y)(4xy) = \ ?
$$

#### Step 1: Multiply the coefficients.
$$
4 \times 4 = 16
$$

#### Step 2: Add the exponents of the same variables.
- For \( x \): The exponents are \( 3 \) and \( 1 \). So, \( x^3 \cdot x^1 = x^{3+1} = x^4 \).
- For \( y \): The exponents are \( 1 \) and \( 1 \). So, \( y^1 \cdot y^1 = y^{1+1} = y^2 \).

#### Final Answer:
$$
(4x^3y)(4xy) = 16x^4y^2
$$

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Problem Solutions:

#### 1. \( (3)(4x) = \ ? \)
- Coefficients: \( 3 \times 4 = 12 \)
- Variables: \( x \) has an exponent of \( 1 \) (since \( x = x^1 \)).
- Final Answer: \( 12x \)

#### 2. \( (5xy)(2x^2y) = \ ? \)
- Coefficients: \( 5 \times 2 = 10 \)
- For \( x \): \( x^1 \cdot x^2 = x^{1+2} = x^3 \)
- For \( y \): \( y^1 \cdot y^1 = y^{1+1} = y^2 \)
- Final Answer: \( 10x^3y^2 \)

#### 3. \( (xy)(7xy) = \ ? \)
- Coefficients: \( 1 \times 7 = 7 \)
- For \( x \): \( x^1 \cdot x^1 = x^{1+1} = x^2 \)
- For \( y \): \( y^1 \cdot y^1 = y^{1+1} = y^2 \)
- Final Answer: \( 7x^2y^2 \)

#### 4. \( (y)(y^3) = \ ? \)
- Coefficients: \( 1 \times 1 = 1 \)
- For \( y \): \( y^1 \cdot y^3 = y^{1+3} = y^4 \)
- Final Answer: \( y^4 \)

#### 5. \( (3x)(4x^2y) = \ ? \)
- Coefficients: \( 3 \times 4 = 12 \)
- For \( x \): \( x^1 \cdot x^2 = x^{1+2} = x^3 \)
- For \( y \): \( y^0 \cdot y^1 = y^{0+1} = y^1 \) (since \( x \) has no \( y \), it's \( y^0 \))
- Final Answer: \( 12x^3y \)

#### 6. \( (6xy)(2x^2y^3) = \ ? \)
- Coefficients: \( 6 \times 2 = 12 \)
- For \( x \): \( x^1 \cdot x^2 = x^{1+2} = x^3 \)
- For \( y \): \( y^1 \cdot y^3 = y^{1+3} = y^4 \)
- Final Answer: \( 12x^3y^4 \)

#### 7. \( (x^2y)(xy) = \ ? \)
- Coefficients: \( 1 \times 1 = 1 \)
- For \( x \): \( x^2 \cdot x^1 = x^{2+1} = x^3 \)
- For \( y \): \( y^1 \cdot y^1 = y^{1+1} = y^2 \)
- Final Answer: \( x^3y^2 \)

#### 8. \( (2xy)(8x^2y) = \ ? \)
- Coefficients: \( 2 \times 8 = 16 \)
- For \( x \): \( x^1 \cdot x^2 = x^{1+2} = x^3 \)
- For \( y \): \( y^1 \cdot y^1 = y^{1+1} = y^2 \)
- Final Answer: \( 16x^3y^2 \)

#### 9. \( (10xy)(2x) = \ ? \)
- Coefficients: \( 10 \times 2 = 20 \)
- For \( x \): \( x^1 \cdot x^1 = x^{1+1} = x^2 \)
- For \( y \): \( y^1 \cdot y^0 = y^{1+0} = y^1 \) (since \( 2x \) has no \( y \), it's \( y^0 \))
- Final Answer: \( 20x^2y \)

#### 10. \( (7x)(2xy) = \ ? \)
- Coefficients: \( 7 \times 2 = 14 \)
- For \( x \): \( x^1 \cdot x^1 = x^{1+1} = x^2 \)
- For \( y \): \( y^0 \cdot y^1 = y^{0+1} = y^1 \) (since \( 7x \) has no \( y \), it's \( y^0 \))
- Final Answer: \( 14x^2y \)

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Final Answers:

1. \( 12x \)
2. \( 10x^3y^2 \)
3. \( 7x^2y^2 \)
4. \( y^4 \)
5. \( 12x^3y \)
6. \( 12x^3y^4 \)
7. \( x^3y^2 \)
8. \( 16x^3y^2 \)
9. \( 20x^2y \)
10. \( 14x^2y \)

\boxed{
\begin{aligned}
1. & \ 12x \\
2. & \ 10x^3y^2 \\
3. & \ 7x^2y^2 \\
4. & \ y^4 \\
5. & \ 12x^3y \\
6. & \ 12x^3y^4 \\
7. & \ x^3y^2 \\
8. & \ 16x^3y^2 \\
9. & \ 20x^2y \\
10. & \ 14x^2y
\end{aligned}
}