Worksheet for simplifying monomial multiplication expressions.

Problem: Simplify each expression by multiplying monomials.

The task involves simplifying expressions that are products of monomials. To do this, we follow these steps:
1. Multiply the coefficients (numerical parts).
2. Add the exponents of like bases (variables).

Let's solve each problem step by step.

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1) \( -1x^2y^3z \times 2x \)

- Coefficients: \( -1 \times 2 = -2 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^3 \) (no other \( y \) term, so it remains \( y^3 \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( -2x^3y^3z \)

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2) \( 2xy \times (-4z) \)

- Coefficients: \( 2 \times (-4) = -8 \)
- Variables:
- \( x \): \( x \) (no other \( x \) term, so it remains \( x \))
- \( y \): \( y \) (no other \( y \) term, so it remains \( y \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( -8xyz \)

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3) \( -8xy \times (-3z) \)

- Coefficients: \( -8 \times (-3) = 24 \)
- Variables:
- \( x \): \( x \) (no other \( x \) term, so it remains \( x \))
- \( y \): \( y \) (no other \( y \) term, so it remains \( y \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( 24xyz \)

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4) \( 6x^2y^3z \times 7x \)

- Coefficients: \( 6 \times 7 = 42 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^3 \) (no other \( y \) term, so it remains \( y^3 \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( 42x^3y^3z \)

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5) \( -10x^2y^3z \times 4x \)

- Coefficients: \( -10 \times 4 = -40 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^3 \) (no other \( y \) term, so it remains \( y^3 \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( -40x^3y^3z \)

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6) \( -2x^2y^2z \times 3xz^2 \)

- Coefficients: \( -2 \times 3 = -6 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^2 \) (no other \( y \) term, so it remains \( y^2 \))
- \( z \): \( z \times z^2 = z^{1+2} = z^3 \)

Result: \( -6x^3y^2z^3 \)

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7) \( 6xy \times (-3z) \)

- Coefficients: \( 6 \times (-3) = -18 \)
- Variables:
- \( x \): \( x \) (no other \( x \) term, so it remains \( x \))
- \( y \): \( y \) (no other \( y \) term, so it remains \( y \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( -18xyz \)

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8) \( -1xy \times (-4z) \)

- Coefficients: \( -1 \times (-4) = 4 \)
- Variables:
- \( x \): \( x \) (no other \( x \) term, so it remains \( x \))
- \( y \): \( y \) (no other \( y \) term, so it remains \( y \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( 4xyz \)

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9) \( -4x^2y^3z \times 7x \)

- Coefficients: \( -4 \times 7 = -28 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^3 \) (no other \( y \) term, so it remains \( y^3 \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( -28x^3y^3z \)

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10) \( 8x^2y^3z \times 5x \)

- Coefficients: \( 8 \times 5 = 40 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^3 \) (no other \( y \) term, so it remains \( y^3 \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( 40x^3y^3z \)

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11) \( -8x^2y^3z \times 3x \)

- Coefficients: \( -8 \times 3 = -24 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^3 \) (no other \( y \) term, so it remains \( y^3 \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( -24x^3y^3z \)

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12) \( -3xy \times 3x^2y \)

- Coefficients: \( -3 \times 3 = -9 \)
- Variables:
- \( x \): \( x \times x^2 = x^{1+2} = x^3 \)
- \( y \): \( y \times y = y^{1+1} = y^2 \)

Result: \( -9x^3y^2 \)

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13) \( 8xy \times (-2z) \)

- Coefficients: \( 8 \times (-2) = -16 \)
- Variables:
- \( x \): \( x \) (no other \( x \) term, so it remains \( x \))
- \( y \): \( y \) (no other \( y \) term, so it remains \( y \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( -16xyz \)

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14) \( 4xy \times (-4z) \)

- Coefficients: \( 4 \times (-4) = -16 \)
- Variables:
- \( x \): \( x \) (no other \( x \) term, so it remains \( x \))
- \( y \): \( y \) (no other \( y \) term, so it remains \( y \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( -16xyz \)

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15) \( 3x^2y^3z \times 3x \)

- Coefficients: \( 3 \times 3 = 9 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^3 \) (no other \( y \) term, so it remains \( y^3 \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( 9x^3y^3z \)

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16) \( 10x^2y^3z \times 3x \)

- Coefficients: \( 10 \times 3 = 30 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^3 \) (no other \( y \) term, so it remains \( y^3 \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( 30x^3y^3z \)

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17) \( 7xy \times 2x^2y \)

- Coefficients: \( 7 \times 2 = 14 \)
- Variables:
- \( x \): \( x \times x^2 = x^{1+2} = x^3 \)
- \( y \): \( y \times y = y^{1+1} = y^2 \)

Result: \( 14x^3y^2 \)

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18) \( 10xy \times (-2z) \)

- Coefficients: \( 10 \times (-2) = -20 \)
- Variables:
- \( x \): \( x \) (no other \( x \) term, so it remains \( x \))
- \( y \): \( y \) (no other \( y \) term, so it remains \( y \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( -20xyz \)

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19) \( -8xy \times 4x^2y \)

- Coefficients: \( -8 \times 4 = -32 \)
- Variables:
- \( x \): \( x \times x^2 = x^{1+2} = x^3 \)
- \( y \): \( y \times y = y^{1+1} = y^2 \)

Result: \( -32x^3y^2 \)

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20) \( 8x^2y^2z \times 7xz^2 \)

- Coefficients: \( 8 \times 7 = 56 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^2 \) (no other \( y \) term, so it remains \( y^2 \))
- \( z \): \( z \times z^2 = z^{1+2} = z^3 \)

Result: \( 56x^3y^2z^3 \)

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21) \( 6x^2y^2z \times 3xz^2 \)

- Coefficients: \( 6 \times 3 = 18 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^2 \) (no other \( y \) term, so it remains \( y^2 \))
- \( z \): \( z \times z^2 = z^{1+2} = z^3 \)

Result: \( 18x^3y^2z^3 \)

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22) \( 2x^2y^3z \times 3x \)

- Coefficients: \( 2 \times 3 = 6 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^3 \) (no other \( y \) term, so it remains \( y^3 \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( 6x^3y^3z \)

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23) \( -4x^2y^2z \times 3xz^2 \)

- Coefficients: \( -4 \times 3 = -12 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^2 \) (no other \( y \) term, so it remains \( y^2 \))
- \( z \): \( z \times z^2 = z^{1+2} = z^3 \)

Result: \( -12x^3y^2z^3 \)

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24) \( -6x^2y^3z \times 2x \)

- Coefficients: \( -6 \times 2 = -12 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^3 \) (no other \( y \) term, so it remains \( y^3 \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( -12x^3y^3z \)

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25) \( -5x^2y^3z \times 7x \)

- Coefficients: \( -5 \times 7 = -35 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^3 \) (no other \( y \) term, so it remains \( y^3 \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( -35x^3y^3z \)

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26) \( -7x^2y^3z \times 2x \)

- Coefficients: \( -7 \times 2 = -14 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^3 \) (no other \( y \) term, so it remains \( y^3 \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( -14x^3y^3z \)

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27) \( 8xy \times 2x^2y \)

- Coefficients: \( 8 \times 2 = 16 \)
- Variables:
- \( x \): \( x \times x^2 = x^{1+2} = x^3 \)
- \( y \): \( y \times y = y^{1+1} = y^2 \)

Result: \( 16x^3y^2 \)

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28) \( 7x^2y^3z \times 5x \)

- Coefficients: \( 7 \times 5 = 35 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^3 \) (no other \( y \) term, so it remains \( y^3 \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( 35x^3y^3z \)

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29) \( -1x^2y^2z \times 4xz^2 \)

- Coefficients: \( -1 \times 4 = -4 \)
- Variables:
- \( x \): \( x^2 \times x = x^{2+1} = x^3 \)
- \( y \): \( y^2 \) (no other \( y \) term, so it remains \( y^2 \))
- \( z \): \( z \times z^2 = z^{1+2} = z^3 \)

Result: \( -4x^3y^2z^3 \)

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30) \( 1xy \times (-2z) \)

- Coefficients: \( 1 \times (-2) = -2 \)
- Variables:
- \( x \): \( x \) (no other \( x \) term, so it remains \( x \))
- \( y \): \( y \) (no other \( y \) term, so it remains \( y \))
- \( z \): \( z \) (no other \( z \) term, so it remains \( z \))

Result: \( -2xyz \)

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Final Answers:

\[
\boxed{
\begin{aligned}
1) & \ -2x^3y^3z \\
2) & \ -8xyz \\
3) & \ 24xyz \\
4) & \ 42x^3y^3z \\
5) & \ -40x^3y^3z \\
6) & \ -6x^3y^2z^3 \\
7) & \ -18xyz \\
8) & \ 4xyz \\
9) & \ -28x^3y^3z \\
10) & \ 40x^3y^3z \\
11) & \ -24x^3y^3z \\
12) & \ -9x^3y^2 \\
13) & \ -16xyz \\
14) & \ -16xyz \\
15) & \ 9x^3y^3z \\
16) & \ 30x^3y^3z \\
17) & \ 14x^3y^2 \\
18) & \ -20xyz \\
19) & \ -32x^3y^2 \\
20) & \ 56x^3y^2z^3 \\
21) & \ 18x^3y^2z^3 \\
22) & \ 6x^3y^3z \\
23) & \ -12x^3y^2z^3 \\
24) & \ -12x^3y^3z \\
25) & \ -35x^3y^3z \\
26) & \ -14x^3y^3z \\
27) & \ 16x^3y^2 \\
28) & \ 35x^3y^3z \\
29) & \ -4x^3y^2z^3 \\
30) & \ -2xyz \\
\end{aligned}
}
\]