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Sagot :
Sure, let's break down the simplification of each of the expressions step by step.
### 12) Simplify [tex]\( 2 \alpha^2 \cdot 4 \alpha \)[/tex]
To simplify the product:
1. Multiply the numerical coefficients: [tex]\(2 \times 4 = 8\)[/tex].
2. Combine the like terms for [tex]\( \alpha \)[/tex]: [tex]\( \alpha^2 \cdot \alpha = \alpha^{2+1} = \alpha^3 \)[/tex].
So, the simplified expression is:
[tex]\[ 2 \alpha^2 \cdot 4 \alpha = 8 \alpha^3 \][/tex]
### 13) Simplify [tex]\((\alpha^2 y)(\alpha y^3)\)[/tex]
To simplify the product:
1. Combine the [tex]\( \alpha \)[/tex]-terms: [tex]\( \alpha^2 \cdot \alpha = \alpha^{2+1} = \alpha^3 \)[/tex].
2. Combine the [tex]\( y \)[/tex]-terms: [tex]\( y \cdot y^3 = y^{1+3} = y^4 \)[/tex].
So, the simplified expression is:
[tex]\[ (\alpha^2 y)(\alpha y^3) = \alpha^3 y^4 \][/tex]
### 15) Simplify [tex]\((2 \alpha \cdot \alpha \cdot 3x \cdot 4 \alpha)\)[/tex]
To simplify the product:
1. Multiply the numerical coefficients: [tex]\(2 \times 3 \times 4 = 24\)[/tex].
2. Combine the like terms for [tex]\( \alpha \)[/tex]: [tex]\( \alpha \cdot \alpha \cdot \alpha = \alpha^{1+1+1} = \alpha^3 \)[/tex].
3. Include [tex]\( x \)[/tex] as it is.
So, the simplified expression is:
[tex]\[ (2 \alpha \cdot \alpha \cdot 3x \cdot 4 \alpha) = 24 \alpha^3 x \][/tex]
### 21) Simplify [tex]\( 2 \alpha (3 x^2 - 4 y) \)[/tex]
To simplify the product inside the brackets:
1. Distribute [tex]\( 2 \alpha \)[/tex] to each term within the parentheses:
- [tex]\( 2 \alpha \cdot 3 x^2 = 6 \alpha x^2 \)[/tex]
- [tex]\( 2 \alpha \cdot -4 y = -8 \alpha y \)[/tex]
So, the simplified expression is:
[tex]\[ 2 \alpha (3 x^2 - 4 y) = 2 \alpha (3 x^2 - 4 y) \][/tex]
### 28) Simplify [tex]\(-4 \alpha (3 \alpha^2 y + 5 x y^2)\)[/tex]
To simplify the product inside the brackets:
1. Distribute [tex]\( -4 \alpha \)[/tex] to each term within the parentheses:
- [tex]\( -4 \alpha \cdot 3 \alpha^2 y = -12 \alpha^3 y \)[/tex]
- [tex]\( -4 \alpha \cdot 5 x y^2 = -20 \alpha x y^2 \)[/tex]
So, the simplified expression is:
[tex]\[ -4 \alpha (3 \alpha^2 y + 5 x y^2) = -4 \alpha (3 \alpha^2 y + 5 x y^2) \][/tex]
### 25) Simplify [tex]\(-2 \alpha (5 x^2 - 8 x y)\)[/tex]
To simplify the product inside the brackets:
1. Distribute [tex]\( -2 \alpha \)[/tex] to each term within the parentheses:
- [tex]\( -2 \alpha \cdot 5 x^2 = -10 \alpha x^2 \)[/tex]
- [tex]\( -2 \alpha \cdot -8 x y = 16 \alpha x y \)[/tex]
So, the simplified expression is:
[tex]\[ -2 \alpha (5 x^2 - 8 x y) = -2 \alpha (5 x^2 - 8 x y) \][/tex]
### 241) Simplify [tex]\(\frac{1}{3} \alpha (16 x^2 - 14 + 2)\)[/tex]
To simplify the product inside the brackets:
1. Simplify inside the parentheses: [tex]\( 16 x^2 - 14 + 2 = 16 x^2 - 12 \)[/tex]
2. Distribute [tex]\(\frac{1}{3} \alpha \)[/tex] to each term:
- [tex]\(\frac{1}{3} \alpha \cdot 16 x^2 = \frac{16}{3} \alpha x^2 \)[/tex]
- [tex]\(\frac{1}{3} \alpha \cdot -12 = -4 \alpha \)[/tex]
So, the simplified expression is:
[tex]\[ \frac{1}{3} \alpha (16 x^2 - 14 + 2) = \frac{1}{3} \alpha (16 x^2 - 12) \][/tex]
### 12) Simplify [tex]\( 2 \alpha^2 \cdot 4 \alpha \)[/tex]
To simplify the product:
1. Multiply the numerical coefficients: [tex]\(2 \times 4 = 8\)[/tex].
2. Combine the like terms for [tex]\( \alpha \)[/tex]: [tex]\( \alpha^2 \cdot \alpha = \alpha^{2+1} = \alpha^3 \)[/tex].
So, the simplified expression is:
[tex]\[ 2 \alpha^2 \cdot 4 \alpha = 8 \alpha^3 \][/tex]
### 13) Simplify [tex]\((\alpha^2 y)(\alpha y^3)\)[/tex]
To simplify the product:
1. Combine the [tex]\( \alpha \)[/tex]-terms: [tex]\( \alpha^2 \cdot \alpha = \alpha^{2+1} = \alpha^3 \)[/tex].
2. Combine the [tex]\( y \)[/tex]-terms: [tex]\( y \cdot y^3 = y^{1+3} = y^4 \)[/tex].
So, the simplified expression is:
[tex]\[ (\alpha^2 y)(\alpha y^3) = \alpha^3 y^4 \][/tex]
### 15) Simplify [tex]\((2 \alpha \cdot \alpha \cdot 3x \cdot 4 \alpha)\)[/tex]
To simplify the product:
1. Multiply the numerical coefficients: [tex]\(2 \times 3 \times 4 = 24\)[/tex].
2. Combine the like terms for [tex]\( \alpha \)[/tex]: [tex]\( \alpha \cdot \alpha \cdot \alpha = \alpha^{1+1+1} = \alpha^3 \)[/tex].
3. Include [tex]\( x \)[/tex] as it is.
So, the simplified expression is:
[tex]\[ (2 \alpha \cdot \alpha \cdot 3x \cdot 4 \alpha) = 24 \alpha^3 x \][/tex]
### 21) Simplify [tex]\( 2 \alpha (3 x^2 - 4 y) \)[/tex]
To simplify the product inside the brackets:
1. Distribute [tex]\( 2 \alpha \)[/tex] to each term within the parentheses:
- [tex]\( 2 \alpha \cdot 3 x^2 = 6 \alpha x^2 \)[/tex]
- [tex]\( 2 \alpha \cdot -4 y = -8 \alpha y \)[/tex]
So, the simplified expression is:
[tex]\[ 2 \alpha (3 x^2 - 4 y) = 2 \alpha (3 x^2 - 4 y) \][/tex]
### 28) Simplify [tex]\(-4 \alpha (3 \alpha^2 y + 5 x y^2)\)[/tex]
To simplify the product inside the brackets:
1. Distribute [tex]\( -4 \alpha \)[/tex] to each term within the parentheses:
- [tex]\( -4 \alpha \cdot 3 \alpha^2 y = -12 \alpha^3 y \)[/tex]
- [tex]\( -4 \alpha \cdot 5 x y^2 = -20 \alpha x y^2 \)[/tex]
So, the simplified expression is:
[tex]\[ -4 \alpha (3 \alpha^2 y + 5 x y^2) = -4 \alpha (3 \alpha^2 y + 5 x y^2) \][/tex]
### 25) Simplify [tex]\(-2 \alpha (5 x^2 - 8 x y)\)[/tex]
To simplify the product inside the brackets:
1. Distribute [tex]\( -2 \alpha \)[/tex] to each term within the parentheses:
- [tex]\( -2 \alpha \cdot 5 x^2 = -10 \alpha x^2 \)[/tex]
- [tex]\( -2 \alpha \cdot -8 x y = 16 \alpha x y \)[/tex]
So, the simplified expression is:
[tex]\[ -2 \alpha (5 x^2 - 8 x y) = -2 \alpha (5 x^2 - 8 x y) \][/tex]
### 241) Simplify [tex]\(\frac{1}{3} \alpha (16 x^2 - 14 + 2)\)[/tex]
To simplify the product inside the brackets:
1. Simplify inside the parentheses: [tex]\( 16 x^2 - 14 + 2 = 16 x^2 - 12 \)[/tex]
2. Distribute [tex]\(\frac{1}{3} \alpha \)[/tex] to each term:
- [tex]\(\frac{1}{3} \alpha \cdot 16 x^2 = \frac{16}{3} \alpha x^2 \)[/tex]
- [tex]\(\frac{1}{3} \alpha \cdot -12 = -4 \alpha \)[/tex]
So, the simplified expression is:
[tex]\[ \frac{1}{3} \alpha (16 x^2 - 14 + 2) = \frac{1}{3} \alpha (16 x^2 - 12) \][/tex]
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