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

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Simplify the expression by following steps:
- step0: Solution:
  $$\frac{x}{2}-\frac{y}{3}+1$$
- step1: Reduce fractions to a common denominator:
  $$\frac{x\times 3}{2\times 3}-\frac{y\times 2}{3\times 2}+\frac{2\times 3}{2\times 3}$$
- step2: Multiply the numbers:
  $$\frac{x\times 3}{6}-\frac{y\times 2}{3\times 2}+\frac{2\times 3}{2\times 3}$$
- step3: Multiply the numbers:
  $$\frac{x\times 3}{6}-\frac{y\times 2}{6}+\frac{2\times 3}{2\times 3}$$
- step4: Multiply the numbers:
  $$\frac{x\times 3}{6}-\frac{y\times 2}{6}+\frac{2\times 3}{6}$$
- step5: Transform the expression:
  $$\frac{x\times 3-y\times 2+2\times 3}{6}$$
- step6: Reorder the terms:
  $$\frac{3x-y\times 2+2\times 3}{6}$$
- step7: Reorder the terms:
  $$\frac{3x-2y+2\times 3}{6}$$
- step8: Multiply the numbers:
  $$\frac{3x-2y+6}{6}$$
Calculate or simplify the expression $$ \frac{3}{x^{3}}-\frac{2}{x^{2}}-\frac{1}{x} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{3}{x^{3}}-\frac{2}{x^{2}}-\frac{1}{x}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{3}{x^{3}}-\frac{2x}{x^{2}\times x}-\frac{x\times x}{x\times x\times x}$$
- step2: Multiply:
  $$\frac{3}{x^{3}}-\frac{2x}{x^{3}}-\frac{x\times x}{x\times x\times x}$$
- step3: Multiply:
  $$\frac{3}{x^{3}}-\frac{2x}{x^{3}}-\frac{x\times x}{x^{3}}$$
- step4: Transform the expression:
  $$\frac{3-2x-x\times x}{x^{3}}$$
- step5: Multiply the terms:
  $$\frac{3-2x-x^{2}}{x^{3}}$$
Calculate or simplify the expression $$ \frac{x-3}{3}-\frac{x-2}{2} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{x-3}{3}-\frac{x-2}{2}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{\left(x-3\right)\times 2}{3\times 2}-\frac{\left(x-2\right)\times 3}{2\times 3}$$
- step2: Multiply the numbers:
  $$\frac{\left(x-3\right)\times 2}{6}-\frac{\left(x-2\right)\times 3}{2\times 3}$$
- step3: Multiply the numbers:
  $$\frac{\left(x-3\right)\times 2}{6}-\frac{\left(x-2\right)\times 3}{6}$$
- step4: Transform the expression:
  $$\frac{\left(x-3\right)\times 2-\left(x-2\right)\times 3}{6}$$
- step5: Multiply the terms:
  $$\frac{2x-6-\left(x-2\right)\times 3}{6}$$
- step6: Multiply the terms:
  $$\frac{2x-6-\left(3x-6\right)}{6}$$
- step7: Subtract the terms:
  $$\frac{-x}{6}$$
- step8: Rewrite the fraction:
  $$-\frac{x}{6}$$
Calculate or simplify the expression $$ \frac{32 x^{3} y^{2} t}{8 x^{3} y t^{2}} $$.
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:
  $$\frac{4y}{t}$$
Calculate or simplify the expression $$ \frac{2 x+2 y}{6} \div 2 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{2x+2y}{6}\div 2$$
- step1: Divide the terms:
  $$\frac{x+y}{3}\div 2$$
- step2: Multiply by the reciprocal:
  $$\frac{x+y}{3}\times \frac{1}{2}$$
- step3: Multiply the terms:
  $$\frac{x+y}{3\times 2}$$
- step4: Multiply the terms:
  $$\frac{x+y}{6}$$
Calculate or simplify the expression $$ \frac{3}{2 x}-\frac{2}{3 x}-\frac{1}{4 x} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{3}{2x}-\frac{2}{3x}-\frac{1}{4x}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{3\times 3\times 2}{2x\times 3\times 2}-\frac{2\times 2\times 2}{3x\times 2\times 2}-\frac{3}{4x\times 3}$$
- step2: Multiply the terms:
  $$\frac{3\times 3\times 2}{12x}-\frac{2\times 2\times 2}{3x\times 2\times 2}-\frac{3}{4x\times 3}$$
- step3: Multiply the terms:
  $$\frac{3\times 3\times 2}{12x}-\frac{2\times 2\times 2}{12x}-\frac{3}{4x\times 3}$$
- step4: Multiply the numbers:
  $$\frac{3\times 3\times 2}{12x}-\frac{2\times 2\times 2}{12x}-\frac{3}{12x}$$
- step5: Transform the expression:
  $$\frac{3\times 3\times 2-2\times 2\times 2-3}{12x}$$
- step6: Multiply the terms:
  $$\frac{18-2\times 2\times 2-3}{12x}$$
- step7: Multiply the terms:
  $$\frac{18-8-3}{12x}$$
- step8: Subtract the numbers:
  $$\frac{7}{12x}$$
Here are the simplified results for each expression:

### TEST 1
1. Simplifying $$ \frac{32 x^{3} y^{2} t}{8 x^{3} y t^{2}} $$:
   $$
   \frac{32 x^{3} y^{2} t}{8 x^{3} y t^{2}} = \frac{4y}{t}
   $$

2. Simplifying $$ \frac{2 x+2 y}{6} \div 2 $$:
   $$
   \frac{2 x+2 y}{6} \div 2 = \frac{x+y}{6}
   $$

3. Simplifying $$ \frac{x}{2}-\frac{y}{3}+1 $$:
   $$
   \frac{x}{2}-\frac{y}{3}+1 = \frac{3x-2y+6}{6}
   $$

### TEST 2
1. Simplifying $$ \frac{x-3}{3}-\frac{x-2}{2} $$:
   $$
   \frac{x-3}{3}-\frac{x-2}{2} = -\frac{x}{6}
   $$

2. Simplifying $$ \frac{3}{2 x}-\frac{2}{3 x}-\frac{1}{4 x} $$:
   $$
   \frac{3}{2 x}-\frac{2}{3 x}-\frac{1}{4 x} = \frac{7}{12x}
   $$

3. Simplifying $$ \frac{3}{x^{3}}-\frac{2}{x^{2}}-\frac{1}{x} $$:
   $$
   \frac{3}{x^{3}}-\frac{2}{x^{2}}-\frac{1}{x} = \frac{3-2x-x^{2}}{x^{3}}
   $$

If you have any further questions or need additional assistance, feel free to ask!
Here are the simplified results for each expression: ### TEST 1 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. $$ \frac{x}{2}-\frac{y}{3}+1 = \frac{3x-2y+6}{6} $$ ### TEST 2 1. $$ \frac{x-3}{3}-\frac{x-2}{2} = -\frac{x}{6} $$ 2. $$ \frac{3}{2 x}-\frac{2}{3 x}-\frac{1}{4 x} = \frac{7}{12x} $$ 3. $$ \frac{3}{x^{3}}-\frac{2}{x^{2}}-\frac{1}{x} = \frac{3-2x-x^{2}}{x^{3}} $$

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

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