Algebraic Expressions & Identities (Topic: Monomial & Binomial ... - Free Printable

Problem: Solving the Products of Algebraic Expressions

The task involves finding the products of given algebraic expressions. We will solve each problem step by step.

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#### 1. \( 2a^3 (3a + 5b) \)

Step-by-Step Solution:
- Distribute \( 2a^3 \) to both terms inside the parentheses:
\[
2a^3 \cdot 3a + 2a^3 \cdot 5b
\]
- Simplify each term:
\[
2a^3 \cdot 3a = 6a^{3+1} = 6a^4
\]
\[
2a^3 \cdot 5b = 10a^3b
\]
- Combine the results:
\[
6a^4 + 10a^3b
\]

Answer:
\[
\boxed{6a^4 + 10a^3b}
\]

---

#### 2. \( -11y^2 (3y + 7) \)

Step-by-Step Solution:
- Distribute \( -11y^2 \) to both terms inside the parentheses:
\[
-11y^2 \cdot 3y + (-11y^2) \cdot 7
\]
- Simplify each term:
\[
-11y^2 \cdot 3y = -33y^{2+1} = -33y^3
\]
\[
-11y^2 \cdot 7 = -77y^2
\]
- Combine the results:
\[
-33y^3 - 77y^2
\]

Answer:
\[
\boxed{-33y^3 - 77y^2}
\]

---

#### 3. \( \frac{6x}{5} (x^5 + y^5) \)

Step-by-Step Solution:
- Distribute \( \frac{6x}{5} \) to both terms inside the parentheses:
\[
\frac{6x}{5} \cdot x^5 + \frac{6x}{5} \cdot y^5
\]
- Simplify each term:
\[
\frac{6x}{5} \cdot x^5 = \frac{6x^{1+5}}{5} = \frac{6x^6}{5}
\]
\[
\frac{6x}{5} \cdot y^5 = \frac{6xy^5}{5}
\]
- Combine the results:
\[
\frac{6x^6}{5} + \frac{6xy^5}{5}
\]

Answer:
\[
\boxed{\frac{6x^6}{5} + \frac{6xy^5}{5}}
\]

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#### 4. \( xy (x^3 - y^3) \)

Step-by-Step Solution:
- Distribute \( xy \) to both terms inside the parentheses:
\[
xy \cdot x^3 + xy \cdot (-y^3)
\]
- Simplify each term:
\[
xy \cdot x^3 = x^{1+3}y = x^4y
\]
\[
xy \cdot (-y^3) = -xy^{1+3} = -xy^4
\]
- Combine the results:
\[
x^4y - xy^4
\]

Answer:
\[
\boxed{x^4y - xy^4}
\]

---

#### 5. \( 0.1y (0.1x^5 + 0.1y) \)

Step-by-Step Solution:
- Distribute \( 0.1y \) to both terms inside the parentheses:
\[
0.1y \cdot 0.1x^5 + 0.1y \cdot 0.1y
\]
- Simplify each term:
\[
0.1y \cdot 0.1x^5 = 0.01x^5y
\]
\[
0.1y \cdot 0.1y = 0.01y^2
\]
- Combine the results:
\[
0.01x^5y + 0.01y^2
\]

Answer:
\[
\boxed{0.01x^5y + 0.01y^2}
\]

---

#### 6. \( \left( -\frac{7}{4}ab^2c - \frac{6}{25}a^2b^2c^2 \right) (-50a^2b^2c^2) \)

Step-by-Step Solution:
- Distribute \( -50a^2b^2c^2 \) to both terms inside the parentheses:
\[
\left( -\frac{7}{4}ab^2c \right) \cdot (-50a^2b^2c^2) + \left( -\frac{6}{25}a^2b^2c^2 \right) \cdot (-50a^2b^2c^2)
\]
- Simplify each term:
\[
\left( -\frac{7}{4}ab^2c \right) \cdot (-50a^2b^2c^2) = \frac{7}{4} \cdot 50 \cdot a^{1+2}b^{2+2}c^{1+2} = \frac{350}{4}a^3b^4c^3 = 87.5a^3b^4c^3
\]
\[
\left( -\frac{6}{25}a^2b^2c^2 \right) \cdot (-50a^2b^2c^2) = \frac{6}{25} \cdot 50 \cdot a^{2+2}b^{2+2}c^{2+2} = \frac{300}{25}a^4b^4c^4 = 12a^4b^4c^4
\]
- Combine the results:
\[
87.5a^3b^4c^3 + 12a^4b^4c^4
\]

Answer:
\[
\boxed{87.5a^3b^4c^3 + 12a^4b^4c^4}
\]

---

#### 7. \( -\frac{8}{27}xyz \left( \frac{3}{2}xyz^2 - \frac{9}{4}xy^2z^5 \right) \)

Step-by-Step Solution:
- Distribute \( -\frac{8}{27}xyz \) to both terms inside the parentheses:
\[
-\frac{8}{27}xyz \cdot \frac{3}{2}xyz^2 + \left( -\frac{8}{27}xyz \right) \cdot \left( -\frac{9}{4}xy^2z^5 \right)
\]
- Simplify each term:
\[
-\frac{8}{27}xyz \cdot \frac{3}{2}xyz^2 = -\frac{8 \cdot 3}{27 \cdot 2} \cdot x^{1+1}y^{1+1}z^{1+2} = -\frac{24}{54}x^2y^2z^3 = -\frac{4}{9}x^2y^2z^3
\]
\[
\left( -\frac{8}{27}xyz \right) \cdot \left( -\frac{9}{4}xy^2z^5 \right) = \frac{8 \cdot 9}{27 \cdot 4} \cdot x^{1+1}y^{1+2}z^{1+5} = \frac{72}{108}x^2y^3z^6 = \frac{2}{3}x^2y^3z^6
\]
- Combine the results:
\[
-\frac{4}{9}x^2y^2z^3 + \frac{2}{3}x^2y^3z^6
\]

Answer:
\[
\boxed{-\frac{4}{9}x^2y^2z^3 + \frac{2}{3}x^2y^3z^6}
\]

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#### 8. \( -\frac{4}{27}xyz \left( \frac{9}{2}x^2yz - \frac{3}{4}xyz^2 \right) \)

Step-by-Step Solution:
- Distribute \( -\frac{4}{27}xyz \) to both terms inside the parentheses:
\[
-\frac{4}{27}xyz \cdot \frac{9}{2}x^2yz + \left( -\frac{4}{27}xyz \right) \cdot \left( -\frac{3}{4}xyz^2 \right)
\]
- Simplify each term:
\[
-\frac{4}{27}xyz \cdot \frac{9}{2}x^2yz = -\frac{4 \cdot 9}{27 \cdot 2} \cdot x^{1+2}y^{1+1}z^{1+1} = -\frac{36}{54}x^3y^2z^2 = -\frac{2}{3}x^3y^2z^2
\]
\[
\left( -\frac{4}{27}xyz \right) \cdot \left( -\frac{3}{4}xyz^2 \right) = \frac{4 \cdot 3}{27 \cdot 4} \cdot x^{1+1}y^{1+1}z^{1+2} = \frac{12}{108}x^2y^2z^3 = \frac{1}{9}x^2y^2z^3
\]
- Combine the results:
\[
-\frac{2}{3}x^3y^2z^2 + \frac{1}{9}x^2y^2z^3
\]

Answer:
\[
\boxed{-\frac{2}{3}x^3y^2z^2 + \frac{1}{9}x^2y^2z^3}
\]

---

#### 9. \( 1.5x (10x^2y - 100xy^2) \)

Step-by-Step Solution:
- Distribute \( 1.5x \) to both terms inside the parentheses:
\[
1.5x \cdot 10x^2y + 1.5x \cdot (-100xy^2)
\]
- Simplify each term:
\[
1.5x \cdot 10x^2y = 15x^{1+2}y = 15x^3y
\]
\[
1.5x \cdot (-100xy^2) = -150x^{1+1}y^2 = -150x^2y^2
\]
- Combine the results:
\[
15x^3y - 150x^2y^2
\]

Answer:
\[
\boxed{15x^3y - 150x^2y^2}
\]

---

#### 10. \( 4.1xy (1.1x - y) \)

Step-by-Step Solution:
- Distribute \( 4.1xy \) to both terms inside the parentheses:
\[
4.1xy \cdot 1.1x + 4.1xy \cdot (-y)
\]
- Simplify each term:
\[
4.1xy \cdot 1.1x = 4.51x^{1+1}y = 4.51x^2y
\]
\[
4.1xy \cdot (-y) = -4.1xy^2
\]
- Combine the results:
\[
4.51x^2y - 4.1xy^2
\]

Answer:
\[
\boxed{4.51x^2y - 4.1xy^2}
\]

---

Final Answers:

1. \( \boxed{6a^4 + 10a^3b} \)
2. \( \boxed{-33y^3 - 77y^2} \)
3. \( \boxed{\frac{6x^6}{5} + \frac{6xy^5}{5}} \)
4. \( \boxed{x^4y - xy^4} \)
5. \( \boxed{0.01x^5y + 0.01y^2} \)
6. \( \boxed{87.5a^3b^4c^3 + 12a^4b^4c^4} \)
7. \( \boxed{-\frac{4}{9}x^2y^2z^3 + \frac{2}{3}x^2y^3z^6} \)
8. \( \boxed{-\frac{2}{3}x^3y^2z^2 + \frac{1}{9}x^2y^2z^3} \)
9. \( \boxed{15x^3y - 150x^2y^2} \)
10. \( \boxed{4.51x^2y - 4.1xy^2} \)

Final Boxed Answer:
\[
\boxed{
\begin{aligned}
&1. 6a^4 + 10a^3b \\
&2. -33y^3 - 77y^2 \\
&3. \frac{6x^6}{5} + \frac{6xy^5}{5} \\
&4. x^4y - xy^4 \\
&5. 0.01x^5y + 0.01y^2 \\
&6. 87.5a^3b^4c^3 + 12a^4b^4c^4 \\
&7. -\frac{4}{9}x^2y^2z^3 + \frac{2}{3}x^2y^3z^6 \\
&8. -\frac{2}{3}x^3y^2z^2 + \frac{1}{9}x^2y^2z^3 \\
&9. 15x^3y - 150x^2y^2 \\
&10. 4.51x^2y - 4.1xy^2
\end{aligned}
}
\]