Multiply As Indicated.${ \frac{3}{x+2} \cdot \frac{x-3}{5} }$

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Introduction

Multiplying fractions is a fundamental concept in mathematics that involves multiplying two or more fractions together. In this article, we will explore the process of multiplying fractions, including the rules and steps involved. We will also provide examples and practice problems to help you understand the concept better.

What is Multiplying Fractions?

Multiplying fractions is a process of multiplying two or more fractions together to get a product. This involves multiplying the numerators (the numbers on top) and the denominators (the numbers on the bottom) of each fraction. The resulting product is a new fraction that represents the result of the multiplication.

Rules for Multiplying Fractions

There are several rules to keep in mind when multiplying fractions:

  • Rule 1: When multiplying fractions, multiply the numerators together to get the new numerator.
  • Rule 2: When multiplying fractions, multiply the denominators together to get the new denominator.
  • Rule 3: When multiplying fractions with different signs (positive or negative), the result will be negative if the signs are different.
  • Rule 4: When multiplying fractions with the same sign (both positive or both negative), the result will be positive.

Step-by-Step Guide to Multiplying Fractions

Here's a step-by-step guide to multiplying fractions:

  1. Identify the fractions: Identify the fractions that need to be multiplied together.
  2. Multiply the numerators: Multiply the numerators of each fraction together to get the new numerator.
  3. Multiply the denominators: Multiply the denominators of each fraction together to get the new denominator.
  4. Simplify the fraction: Simplify the resulting fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).
  5. Write the final answer: Write the final answer as a simplified fraction.

Example 1: Multiplying Two Fractions

Let's consider an example of multiplying two fractions:

3x+2⋅x−35{ \frac{3}{x+2} \cdot \frac{x-3}{5} }

To multiply these fractions, we need to follow the steps outlined above:

  1. Identify the fractions: The two fractions are 3x+2{\frac{3}{x+2}} and x−35{\frac{x-3}{5}}.
  2. Multiply the numerators: Multiply the numerators together to get the new numerator: 3⋅(x−3)=3x−9{3 \cdot (x-3) = 3x - 9}.
  3. Multiply the denominators: Multiply the denominators together to get the new denominator: (x+2)â‹…5=5x+10{(x+2) \cdot 5 = 5x + 10}.
  4. Simplify the fraction: Simplify the resulting fraction by dividing both the numerator and the denominator by their GCD.
  5. Write the final answer: The final answer is 3x−95x+10{\frac{3x - 9}{5x + 10}}.

Example 2: Multiplying Three Fractions

Let's consider an example of multiplying three fractions:

2x+1⋅x−23⋅4x−1{ \frac{2}{x+1} \cdot \frac{x-2}{3} \cdot \frac{4}{x-1} }

To multiply these fractions, we need to follow the steps outlined above:

  1. Identify the fractions: The three fractions are 2x+1{\frac{2}{x+1}}, x−23{\frac{x-2}{3}}, and 4x−1{\frac{4}{x-1}}.
  2. Multiply the numerators: Multiply the numerators together to get the new numerator: 2⋅(x−2)⋅4=8x−16{2 \cdot (x-2) \cdot 4 = 8x - 16}.
  3. Multiply the denominators: Multiply the denominators together to get the new denominator: (x+1)⋅3⋅(x−1)=3x2−3{(x+1) \cdot 3 \cdot (x-1) = 3x^2 - 3}.
  4. Simplify the fraction: Simplify the resulting fraction by dividing both the numerator and the denominator by their GCD.
  5. Write the final answer: The final answer is 8x−163x2−3{\frac{8x - 16}{3x^2 - 3}}.

Practice Problems

Here are some practice problems to help you understand the concept of multiplying fractions:

  1. Problem 1: Multiply the fractions 4x+3{\frac{4}{x+3}} and x−22{\frac{x-2}{2}}.
  2. Problem 2: Multiply the fractions 3x−1{\frac{3}{x-1}} and x+24{\frac{x+2}{4}}.
  3. Problem 3: Multiply the fractions 2x+2{\frac{2}{x+2}} and x−13{\frac{x-1}{3}}.

Conclusion

Introduction

Multiplying fractions is a fundamental concept in mathematics that involves multiplying two or more fractions together. In this article, we will provide a Q&A guide to help you understand the concept of multiplying fractions and address any questions or concerns you may have.

Q: What is the rule for multiplying fractions?

A: The rule for multiplying fractions is to multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator.

Q: How do I multiply fractions with different signs?

A: When multiplying fractions with different signs, the result will be negative if the signs are different. For example, 3x+2⋅x−35{\frac{3}{x+2} \cdot \frac{x-3}{5}} will result in a negative fraction.

Q: How do I multiply fractions with the same sign?

A: When multiplying fractions with the same sign, the result will be positive. For example, 3x+2⋅x−35{\frac{3}{x+2} \cdot \frac{x-3}{5}} will result in a positive fraction.

Q: What is the greatest common divisor (GCD) and how do I use it to simplify fractions?

A: The GCD is the largest number that divides both the numerator and the denominator of a fraction. To simplify a fraction, divide both the numerator and the denominator by their GCD.

Q: How do I multiply fractions with variables?

A: When multiplying fractions with variables, multiply the variables together and simplify the resulting fraction. For example, 2xx+2⋅x−23{\frac{2x}{x+2} \cdot \frac{x-2}{3}} will result in 2x(x−2)3(x+2){\frac{2x(x-2)}{3(x+2)}}.

Q: Can I multiply fractions with different denominators?

A: Yes, you can multiply fractions with different denominators. To do this, multiply the numerators together and multiply the denominators together.

Q: How do I multiply fractions with negative exponents?

A: When multiplying fractions with negative exponents, multiply the numerators together and multiply the denominators together. For example, 2x+2⋅x−35−1{\frac{2}{x+2} \cdot \frac{x-3}{5^{-1}}} will result in 2(x−3)5−1(x+2){\frac{2(x-3)}{5^{-1}(x+2)}}.

Q: Can I multiply fractions with complex numbers?

A: Yes, you can multiply fractions with complex numbers. To do this, multiply the numerators together and multiply the denominators together.

Q: How do I multiply fractions with decimals?

A: When multiplying fractions with decimals, convert the decimals to fractions and then multiply the fractions together.

Q: Can I multiply fractions with mixed numbers?

A: Yes, you can multiply fractions with mixed numbers. To do this, convert the mixed numbers to improper fractions and then multiply the fractions together.

Q: How do I multiply fractions with zero?

A: When multiplying fractions with zero, the result will be zero. For example, 3x+2â‹…05{\frac{3}{x+2} \cdot \frac{0}{5}} will result in zero.

Conclusion

Multiplying fractions is a fundamental concept in mathematics that involves multiplying two or more fractions together. By following the rules and steps outlined in this Q&A guide, you can become proficient in multiplying fractions and apply it to a wide range of mathematical problems. Remember to identify the fractions, multiply the numerators and denominators, simplify the fraction, and write the final answer as a simplified fraction. With practice and patience, you can master the concept of multiplying fractions and become a confident mathematician.