Multiplying Monomials Worksheets - Free Printable

To solve the problems involving multiplying monomials and binomials, we need to apply the rules of exponents and the distributive property. Let's go through each problem step by step.

---

Problem 1:

$$
-2v^3 \cdot 5v^6
$$

#### Solution:
1. Multiply the coefficients: $-2 \cdot 5 = -10$.
2. Add the exponents of the like bases (using the rule $a^m \cdot a^n = a^{m+n}$):
$$
v^3 \cdot v^6 = v^{3+6} = v^9
$$
3. Combine the results:
$$
-2v^3 \cdot 5v^6 = -10v^9
$$

#### Final Answer:
$$
\boxed{-10v^9}
$$

---

Problem 2:

$$
\frac{1}{4}x^2y^3 \cdot (-8xy^2)
$$

#### Solution:
1. Multiply the coefficients: $\frac{1}{4} \cdot (-8) = -2$.
2. Add the exponents of the like bases:
- For $x$: $x^2 \cdot x = x^{2+1} = x^3$.
- For $y$: $y^3 \cdot y^2 = y^{3+2} = y^5$.
3. Combine the results:
$$
\frac{1}{4}x^2y^3 \cdot (-8xy^2) = -2x^3y^5
$$

#### Final Answer:
$$
\boxed{-2x^3y^5}
$$

---

Problem 3:

$$
-4a^2b^3 \cdot 7ab^5
$$

#### Solution:
1. Multiply the coefficients: $-4 \cdot 7 = -28$.
2. Add the exponents of the like bases:
- For $a$: $a^2 \cdot a = a^{2+1} = a^3$.
- For $b$: $b^3 \cdot b^5 = b^{3+5} = b^8$.
3. Combine the results:
$$
-4a^2b^3 \cdot 7ab^5 = -28a^3b^8
$$

#### Final Answer:
$$
\boxed{-28a^3b^8}
$$

---

Problem 4:

$$
(3x - 2)(x + 5)
$$

#### Solution:
Use the distributive property (also known as the FOIL method for binomials):
1. Distribute each term in the first binomial to each term in the second binomial:
$$
(3x - 2)(x + 5) = 3x \cdot x + 3x \cdot 5 - 2 \cdot x - 2 \cdot 5
$$
2. Perform the multiplications:
$$
3x \cdot x = 3x^2, \quad 3x \cdot 5 = 15x, \quad -2 \cdot x = -2x, \quad -2 \cdot 5 = -10
$$
3. Combine like terms:
$$
3x^2 + 15x - 2x - 10 = 3x^2 + 13x - 10
$$

#### Final Answer:
$$
\boxed{3x^2 + 13x - 10}
$$

---

Problem 5:

$$
12n^3m^2 \cdot 4m^3
$$

#### Solution:
1. Multiply the coefficients: $12 \cdot 4 = 48$.
2. Add the exponents of the like bases:
- For $n$: $n^3$ remains $n^3$ (no other $n$ term).
- For $m$: $m^2 \cdot m^3 = m^{2+3} = m^5$.
3. Combine the results:
$$
12n^3m^2 \cdot 4m^3 = 48n^3m^5
$$

#### Final Answer:
$$
\boxed{48n^3m^5}
$$

---

Problem 6:

$$
-2cd^2 \cdot \left(-\frac{3}{4}c^2d^3\right)
$$

#### Solution:
1. Multiply the coefficients: $-2 \cdot \left(-\frac{3}{4}\right) = \frac{6}{4} = \frac{3}{2}$.
2. Add the exponents of the like bases:
- For $c$: $c \cdot c^2 = c^{1+2} = c^3$.
- For $d$: $d^2 \cdot d^3 = d^{2+3} = d^5$.
3. Combine the results:
$$
-2cd^2 \cdot \left(-\frac{3}{4}c^2d^3\right) = \frac{3}{2}c^3d^5
$$

#### Final Answer:
$$
\boxed{\frac{3}{2}c^3d^5}
$$

---

Problem 7:

$$
-\frac{1}{2}w^3 \cdot \left(-\frac{1}{4}w^2\right)
$$

#### Solution:
1. Multiply the coefficients: $-\frac{1}{2} \cdot \left(-\frac{1}{4}\right) = \frac{1}{8}$.
2. Add the exponents of the like bases:
$$
w^3 \cdot w^2 = w^{3+2} = w^5
$$
3. Combine the results:
$$
-\frac{1}{2}w^3 \cdot \left(-\frac{1}{4}w^2\right) = \frac{1}{8}w^5
$$

#### Final Answer:
$$
\boxed{\frac{1}{8}w^5}
$$

---

Problem 8:

$$
3x^2 - 10x^3
$$

#### Solution:
This expression is already simplified. There are no like terms to combine.

#### Final Answer:
$$
\boxed{3x^2 - 10x^3}
$$

---

Final Answers:

1. $\boxed{-10v^9}$
2. $\boxed{-2x^3y^5}$
3. $\boxed{-28a^3b^8}$
4. $\boxed{3x^2 + 13x - 10}$
5. $\boxed{48n^3m^5}$
6. $\boxed{\frac{3}{2}c^3d^5}$
7. $\boxed{\frac{1}{8}w^5}$
8. $\boxed{3x^2 - 10x^3}$

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Boxed Final Answer:
$$
\boxed{
\begin{aligned}
1. & \ -10v^9 \\
2. & \ -2x^3y^5 \\
3. & \ -28a^3b^8 \\
4. & \ 3x^2 + 13x - 10 \\
5. & \ 48n^3m^5 \\
6. & \ \frac{3}{2}c^3d^5 \\
7. & \ \frac{1}{8}w^5 \\
8. & \ 3x^2 - 10x^3
\end{aligned}
}
$$