Solve the following rational expressions: 1. Multiply and simplify $\frac{x}{2} * \frac{3x}{10} * \frac{2}{12}$ 2. Multiply and simplify $\frac{10}{x} * \frac{12}{5x^2}$ 3. Divid... Solve the following rational expressions: 1. Multiply and simplify $\frac{x}{2} * \frac{3x}{10} * \frac{2}{12}$ 2. Multiply and simplify $\frac{10}{x} * \frac{12}{5x^2}$ 3. Divide and simplify $\frac{2x}{3} \div \frac{5x}{6}$
Understand the Problem
The image contains a series of math problems involving rational expressions. The problems involve multiplication and division of algebraic fractions. Each problem requires simplification after performing the operation.
Answer
1. $\frac{x^2}{40}$ 2. $\frac{24}{x^3}$ 3. $\frac{4}{5}$
Answer for screen readers
- $\frac{x^2}{40}$
- $\frac{24}{x^3}$
- $\frac{4}{5}$
Steps to Solve
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Solve the first multiplication problem Multiply the numerators and denominators: $\frac{x}{2} \cdot \frac{3x}{10} \cdot \frac{2}{12} = \frac{x \cdot 3x \cdot 2}{2 \cdot 10 \cdot 12} = \frac{6x^2}{240}$
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Simplify the result Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 6: $\frac{6x^2}{240} = \frac{x^2}{40}$
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Solve the second multiplication problem Multiply the numerators and denominators: $\frac{10}{x} \cdot \frac{12}{5x^2} = \frac{10 \cdot 12}{x \cdot 5x^2} = \frac{120}{5x^3}$
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Simplify the result Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 5: $\frac{120}{5x^3} = \frac{24}{x^3}$
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Solve the division problem To divide fractions, multiply by the reciprocal of the second fraction: $\frac{2x}{3} \div \frac{5x}{6} = \frac{2x}{3} \cdot \frac{6}{5x} = \frac{2x \cdot 6}{3 \cdot 5x} = \frac{12x}{15x}$
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Simplify the result Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is $3x$: $\frac{12x}{15x} = \frac{4}{5}$
- $\frac{x^2}{40}$
- $\frac{24}{x^3}$
- $\frac{4}{5}$
More Information
Rational expressions are like fractions but with variables. The key to solving these problems is to remember how to multiply and divide fractions, then simplify.
Tips
A common mistake is forgetting to simplify the fractions after multiplying or dividing. Another mistake is not correctly finding the reciprocal when dividing fractions.
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