Multiply. Assume All Denominators Are Nonzero. Simplify Your Answer. − 12 X 3 4 Y 3 ⋅ 8 Y 2 6 X 6 -\frac{12 X^3}{4 Y^3} \cdot \frac{8 Y^2}{6 X^6} − 4 Y 3 12 X 3 ​ ⋅ 6 X 6 8 Y 2 ​

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Introduction

Multiplying rational expressions is a fundamental concept in algebra that involves multiplying two or more fractions together. In this article, we will explore the process of multiplying rational expressions, with a focus on simplifying the resulting expression. We will use the given problem, −12x34y3⋅8y26x6-\frac{12 x^3}{4 y^3} \cdot \frac{8 y^2}{6 x^6}, as a case study to demonstrate the steps involved in multiplying rational expressions.

What are Rational Expressions?

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be added, subtracted, multiplied, and divided, just like regular fractions. However, when multiplying rational expressions, we need to follow specific rules to simplify the resulting expression.

Multiplying Rational Expressions: The Basics

To multiply rational expressions, we follow these basic steps:

  1. Multiply the numerators: Multiply the numerators of the two fractions together.
  2. Multiply the denominators: Multiply the denominators of the two fractions together.
  3. Simplify the resulting expression: Simplify the resulting expression by canceling out any common factors between the numerator and denominator.

Multiplying the Numerators and Denominators

Let's apply these steps to the given problem:

−12x34y3⋅8y26x6-\frac{12 x^3}{4 y^3} \cdot \frac{8 y^2}{6 x^6}

Step 1: Multiply the numerators

12x3⋅8y2=96x3y212 x^3 \cdot 8 y^2 = 96 x^3 y^2

Step 2: Multiply the denominators

4y3⋅6x6=24x6y34 y^3 \cdot 6 x^6 = 24 x^6 y^3

Step 3: Simplify the resulting expression

Now that we have multiplied the numerators and denominators, we can simplify the resulting expression by canceling out any common factors between the numerator and denominator.

Simplifying the Expression

To simplify the expression, we need to find the greatest common factor (GCF) of the numerator and denominator. The GCF of 96x3y296 x^3 y^2 and 24x6y324 x^6 y^3 is 24x3y224 x^3 y^2.

We can now cancel out the GCF from both the numerator and denominator:

96x3y224x6y3=4x3y\frac{96 x^3 y^2}{24 x^6 y^3} = \frac{4}{x^3 y}

Conclusion

Multiplying rational expressions involves multiplying the numerators and denominators together and then simplifying the resulting expression by canceling out any common factors between the numerator and denominator. By following these steps, we can simplify complex rational expressions and arrive at a more manageable form.

Example Problems

Here are a few example problems to help you practice multiplying rational expressions:

  1. 2x23y2⋅4y25x4\frac{2 x^2}{3 y^2} \cdot \frac{4 y^2}{5 x^4}
  2. 3x32y3⋅2y34x6\frac{3 x^3}{2 y^3} \cdot \frac{2 y^3}{4 x^6}
  3. 5x22y2⋅3y24x4\frac{5 x^2}{2 y^2} \cdot \frac{3 y^2}{4 x^4}

Tips and Tricks

Here are a few tips and tricks to help you multiply rational expressions:

  • Use the distributive property: When multiplying rational expressions, use the distributive property to multiply the numerators and denominators separately.
  • Simplify the resulting expression: Simplify the resulting expression by canceling out any common factors between the numerator and denominator.
  • Check your work: Double-check your work to ensure that you have simplified the expression correctly.

Introduction

Multiplying rational expressions is a fundamental concept in algebra that involves multiplying two or more fractions together. In this article, we will explore the process of multiplying rational expressions, with a focus on simplifying the resulting expression. We will also provide a Q&A section to help you better understand the concept and address any common questions or concerns.

Q&A: Multiplying Rational Expressions

Q: What is the first step in multiplying rational expressions? A: The first step in multiplying rational expressions is to multiply the numerators together.

Q: How do I multiply the numerators? A: To multiply the numerators, simply multiply the numbers and variables together. For example, if you have the expression 2x23y2⋅4y25x4\frac{2 x^2}{3 y^2} \cdot \frac{4 y^2}{5 x^4}, you would multiply the numerators together as follows: 2x2⋅4y2=8x2y22 x^2 \cdot 4 y^2 = 8 x^2 y^2.

Q: What is the second step in multiplying rational expressions? A: The second step in multiplying rational expressions is to multiply the denominators together.

Q: How do I multiply the denominators? A: To multiply the denominators, simply multiply the numbers and variables together. For example, if you have the expression 2x23y2⋅4y25x4\frac{2 x^2}{3 y^2} \cdot \frac{4 y^2}{5 x^4}, you would multiply the denominators together as follows: 3y2⋅5x4=15x4y23 y^2 \cdot 5 x^4 = 15 x^4 y^2.

Q: What is the third step in multiplying rational expressions? A: The third step in multiplying rational expressions is to simplify the resulting expression by canceling out any common factors between the numerator and denominator.

Q: How do I simplify the resulting expression? A: To simplify the resulting expression, you need to find the greatest common factor (GCF) of the numerator and denominator. The GCF is the largest expression that divides both the numerator and denominator without leaving a remainder. Once you have found the GCF, you can cancel it out from both the numerator and denominator.

Q: What if the GCF is not a single number or variable? A: If the GCF is not a single number or variable, you can still cancel it out from both the numerator and denominator. For example, if the GCF is x2y2x^2 y^2, you can cancel it out as follows: 8x2y215x4y2=815x2\frac{8 x^2 y^2}{15 x^4 y^2} = \frac{8}{15 x^2}.

Q: Can I multiply rational expressions with different variables? A: Yes, you can multiply rational expressions with different variables. When multiplying rational expressions with different variables, you need to multiply the variables together as well as the numbers.

Q: What if I have a rational expression with a negative sign? A: If you have a rational expression with a negative sign, you can simply multiply the expression by -1 to get rid of the negative sign. For example, if you have the expression −2x23y2-\frac{2 x^2}{3 y^2}, you can multiply it by -1 to get rid of the negative sign: −2x23y2⋅−1=2x23y2-\frac{2 x^2}{3 y^2} \cdot -1 = \frac{2 x^2}{3 y^2}.

Conclusion

Multiplying rational expressions involves multiplying the numerators and denominators together and then simplifying the resulting expression by canceling out any common factors between the numerator and denominator. By following these steps, you can simplify complex rational expressions and arrive at a more manageable form.

Practice Problems

Here are a few practice problems to help you practice multiplying rational expressions:

  1. 3x24y2⋅2y25x4\frac{3 x^2}{4 y^2} \cdot \frac{2 y^2}{5 x^4}
  2. 2x33y3⋅3y34x6\frac{2 x^3}{3 y^3} \cdot \frac{3 y^3}{4 x^6}
  3. 4x25y2⋅3y22x4\frac{4 x^2}{5 y^2} \cdot \frac{3 y^2}{2 x^4}

Tips and Tricks

Here are a few tips and tricks to help you multiply rational expressions:

  • Use the distributive property: When multiplying rational expressions, use the distributive property to multiply the numerators and denominators separately.
  • Simplify the resulting expression: Simplify the resulting expression by canceling out any common factors between the numerator and denominator.
  • Check your work: Double-check your work to ensure that you have simplified the expression correctly.

By following these tips and tricks, you can become more confident and proficient in multiplying rational expressions.