What Is The Product Of The Following Expression? 2 Y Y − 3 ⋅ 4 Y − 12 2 Y + 6 \frac{2y}{y-3} \cdot \frac{4y-12}{2y+6} Y − 3 2 Y ​ ⋅ 2 Y + 6 4 Y − 12 ​ A. 2 3 \frac{2}{3} 3 2 ​ B. 10 9 \frac{10}{9} 9 10 ​ C. 4 Y Y − 3 \frac{4y}{y-3} Y − 3 4 Y ​ D. 4 Y Y + 3 \frac{4y}{y+3} Y + 3 4 Y ​

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the product of two fractions. We will use the given expression 2yy−3⋅4y−122y+6\frac{2y}{y-3} \cdot \frac{4y-12}{2y+6} as a case study to demonstrate the steps involved in simplifying algebraic expressions.

Understanding the Expression

Before we dive into the simplification process, let's take a closer look at the given expression. The expression consists of two fractions, each with a numerator and a denominator. The first fraction is 2yy−3\frac{2y}{y-3}, and the second fraction is 4y−122y+6\frac{4y-12}{2y+6}. Our goal is to simplify the product of these two fractions.

Step 1: Factor the Numerators and Denominators

To simplify the expression, we need to factor the numerators and denominators of each fraction. Let's start with the first fraction, 2yy−3\frac{2y}{y-3}. We can factor out a 2 from the numerator, giving us 2(y)y−3\frac{2(y)}{y-3}. The denominator, y−3y-3, is already factored.

Next, let's factor the second fraction, 4y−122y+6\frac{4y-12}{2y+6}. We can factor out a 4 from the numerator, giving us 4(y−3)2y+6\frac{4(y-3)}{2y+6}. The denominator, 2y+62y+6, can be factored as 2(y+3)2(y+3).

Step 2: Simplify the Expression

Now that we have factored the numerators and denominators, we can simplify the expression. We can cancel out common factors between the numerators and denominators.

The first fraction, 2(y)y−3\frac{2(y)}{y-3}, has a common factor of yy between the numerator and denominator. We can cancel out this factor, giving us 2y−3\frac{2}{y-3}.

The second fraction, 4(y−3)2(y+3)\frac{4(y-3)}{2(y+3)}, has a common factor of 22 between the numerator and denominator. We can cancel out this factor, giving us 2(y−3)y+3\frac{2(y-3)}{y+3}.

Step 3: Multiply the Fractions

Now that we have simplified each fraction, we can multiply them together. We can multiply the numerators and denominators separately.

The numerator of the first fraction is 22, and the numerator of the second fraction is 2(y−3)2(y-3). We can multiply these two numerators together, giving us 2⋅2(y−3)=4(y−3)2 \cdot 2(y-3) = 4(y-3).

The denominator of the first fraction is y−3y-3, and the denominator of the second fraction is y+3y+3. We can multiply these two denominators together, giving us (y−3)(y+3)(y-3)(y+3).

Step 4: Simplify the Result

Now that we have multiplied the fractions, we can simplify the result. We can factor the numerator and denominator of the resulting fraction.

The numerator, 4(y−3)4(y-3), can be factored as 4(y−3)4(y-3). The denominator, (y−3)(y+3)(y-3)(y+3), can be factored as (y−3)(y+3)(y-3)(y+3).

Conclusion

In conclusion, the product of the given expression 2yy−3⋅4y−122y+6\frac{2y}{y-3} \cdot \frac{4y-12}{2y+6} is 4(y−3)(y−3)(y+3)\frac{4(y-3)}{(y-3)(y+3)}. We can simplify this expression further by canceling out the common factor of y−3y-3 between the numerator and denominator, giving us 4y+3\frac{4}{y+3}.

Answer

The final answer is 4y+3\boxed{\frac{4}{y+3}}.

Discussion

The given expression 2yy−3⋅4y−122y+6\frac{2y}{y-3} \cdot \frac{4y-12}{2y+6} is a classic example of an algebraic expression that requires simplification. By following the steps outlined in this article, we can simplify the expression and arrive at the final answer.

In this article, we used the distributive property to factor the numerators and denominators of each fraction. We then canceled out common factors between the numerators and denominators to simplify the expression. Finally, we multiplied the fractions together and simplified the result.

The process of simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, you can simplify even the most complex algebraic expressions.

Common Mistakes

When simplifying algebraic expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not factoring the numerators and denominators: Failing to factor the numerators and denominators can make it difficult to simplify the expression.
  • Not canceling out common factors: Failing to cancel out common factors between the numerators and denominators can lead to an incorrect answer.
  • Not multiplying the fractions correctly: Failing to multiply the fractions correctly can lead to an incorrect answer.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

  • Use the distributive property: The distributive property is a powerful tool for factoring numerators and denominators.
  • Cancel out common factors: Canceling out common factors between the numerators and denominators can simplify the expression.
  • Multiply the fractions correctly: Multiplying the fractions correctly is essential for arriving at the correct answer.

Introduction

In our previous article, we explored the process of simplifying algebraic expressions, with a focus on the product of two fractions. We used the given expression 2yy−3⋅4y−122y+6\frac{2y}{y-3} \cdot \frac{4y-12}{2y+6} as a case study to demonstrate the steps involved in simplifying algebraic expressions. In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to factor the numerators and denominators of each fraction. This involves using the distributive property to break down the numerators and denominators into their simplest forms.

Q: How do I factor the numerators and denominators of a fraction?

A: To factor the numerators and denominators of a fraction, you can use the distributive property. For example, if you have a fraction with a numerator of 2x+42x+4 and a denominator of x−2x-2, you can factor the numerator as 2(x+2)2(x+2) and the denominator as (x−2)(x-2).

Q: What is the next step in simplifying an algebraic expression?

A: After factoring the numerators and denominators, the next step is to cancel out common factors between the numerators and denominators. This involves identifying any common factors between the numerator and denominator and canceling them out.

Q: How do I cancel out common factors between the numerators and denominators?

A: To cancel out common factors between the numerators and denominators, you can simply divide both the numerator and denominator by the common factor. For example, if you have a fraction with a numerator of 6x6x and a denominator of 3x3x, you can cancel out the common factor of 3x3x by dividing both the numerator and denominator by 3x3x.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to multiply the fractions together and simplify the result. This involves multiplying the numerators and denominators of each fraction together and simplifying the resulting expression.

Q: How do I multiply fractions together?

A: To multiply fractions together, you can simply multiply the numerators and denominators of each fraction together. For example, if you have two fractions with numerators of 2x2x and 3x3x and denominators of x−2x-2 and x+2x+2, you can multiply the numerators and denominators together to get 6x2(x−2)(x+2)\frac{6x^2}{(x-2)(x+2)}.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Not factoring the numerators and denominators
  • Not canceling out common factors between the numerators and denominators
  • Not multiplying the fractions correctly

Q: What are some tips and tricks for simplifying algebraic expressions?

A: Some tips and tricks for simplifying algebraic expressions include:

  • Using the distributive property to factor the numerators and denominators
  • Canceling out common factors between the numerators and denominators
  • Multiplying the fractions correctly

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article and avoiding common mistakes, you can simplify even the most complex algebraic expressions. Remember to factor the numerators and denominators, cancel out common factors, and multiply the fractions correctly to arrive at the correct answer.

Common Algebraic Expressions

Here are some common algebraic expressions that you may encounter:

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

Practice Problems

Here are some practice problems to help you practice simplifying algebraic expressions:

  • Simplify the expression 2xx−1⋅3x−2x+1\frac{2x}{x-1} \cdot \frac{3x-2}{x+1}.
  • Simplify the expression x+2x−3⋅2x−1x+4\frac{x+2}{x-3} \cdot \frac{2x-1}{x+4}.
  • Simplify the expression 3x−2x+1⋅x+2x−3\frac{3x-2}{x+1} \cdot \frac{x+2}{x-3}.

Answer Key

Here are the answers to the practice problems:

  • 6x2−4xx2−1\frac{6x^2-4x}{x^2-1}
  • 2x2+5x−2x2+3x−12\frac{2x^2+5x-2}{x^2+3x-12}
  • 6x2−4xx2−9\frac{6x^2-4x}{x^2-9}