Section 4 Algebraic fractions TEST 1 Simplify: 1. $$ \frac{32 x^{3} y^{2} t}{8 x^{3} y t^{2}} $$ $$ \rightarrow \frac{2 x+2 y}{6} \div 2 $$ 5. $$ \frac{x}{2}-\frac{y}{3}+1 $$

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To simplify the given algebraic fractions, we will follow the rules of simplifying fractions.

1. Simplify $$ \frac{32 x^{3} y^{2} t}{8 x^{3} y t^{2}} $$:
   - Cancel out common factors in the numerator and denominator.
   - Simplify the remaining expression.

2. Simplify $$ \frac{2 x+2 y}{6} \div 2 $$:
   - Divide the fraction by 2.

3. Simplify $$ \frac{x}{2}-\frac{y}{3}+1 $$:
   - Combine like terms.

Let's simplify each expression step by step.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{32x^{3}y^{2}t}{8x^{3}yt^{2}}$$
- step1: Reduce the fraction:
  $$\frac{32y^{2}t}{8yt^{2}}$$
- step2: Reduce the fraction:
  $$\frac{32y^{2-1}t}{8t^{2}}$$
- step3: Reduce the fraction:
  $$\frac{32yt}{8t^{2}}$$
- step4: Reduce the fraction:
  $$\frac{32y}{8t^{2-1}}$$
- step5: Reduce the fraction:
  $$\frac{32y}{8t}$$
- step6: Divide the terms:
  $$\frac{4y}{t}$$
1. The simplified form of $$ \frac{32 x^{3} y^{2} t}{8 x^{3} y t^{2}} $$ is $$ \frac{4y}{t} $$.

2. Simplify $$ \frac{2 x+2 y}{6} \div 2 $$:
   - Divide the fraction by 2:
   - $$ \frac{2 x+2 y}{6} \div 2 = \frac{2 x+2 y}{6} 	imes \frac{1}{2} = \frac{x+y}{6} $$

3. Simplify $$ \frac{x}{2}-\frac{y}{3}+1 $$:
   - Combine like terms:
   - $$ \frac{x}{2}-\frac{y}{3}+1 = \frac{3x-2y+6}{6} $$

Therefore, the simplified forms are:
1. $$ \frac{32 x^{3} y^{2} t}{8 x^{3} y t^{2}} = \frac{4y}{t} $$
2. $$ \frac{2 x+2 y}{6} \div 2 = \frac{x+y}{6} $$
3. $$ \frac{x}{2}-\frac{y}{3}+1 = \frac{3x-2y+6}{6} $$
1. $$ \frac{32 x^{3} y^{2} t}{8 x^{3} y t^{2}} = \frac{4y}{t} $$ 2. $$ \frac{2 x+2 y}{6} \div 2 = \frac{x+y}{6} $$ 3. $$ \frac{x}{2}-\frac{y}{3}+1 = \frac{3x-2y+6}{6} $$

Section 4 Algebraic fractions TEST 1 Simplify: 1. \( \frac{32 x^{3} y^{2} t}{8 x^{3} y t^{2}} \) \( \rightarrow \frac{2 x+2 y}{6} \div 2 \) 5. \( \frac{x}{2}-\frac{y}{3}+1 \)

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