Multiply The Following Expression And Simplify Your Answer As Much As Possible:$\[ \frac{x^2 + 9xy + 8y^2}{2x - 6y} \cdot \frac{x - 3y}{x^2 + 6xy - 16y^2} \\]

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Introduction

In this article, we will be multiplying the given expression and simplifying our answer as much as possible. The expression given is a product of two rational expressions, and we will be using the rules of algebra to simplify it. We will start by factoring the numerator and denominator of each rational expression, and then we will multiply them together.

Factoring the Numerator and Denominator

To simplify the given expression, we need to factor the numerator and denominator of each rational expression. The numerator of the first rational expression is x2+9xy+8y2x^2 + 9xy + 8y^2, and the denominator is 2xβˆ’6y2x - 6y. The numerator of the second rational expression is xβˆ’3yx - 3y, and the denominator is x2+6xyβˆ’16y2x^2 + 6xy - 16y^2.

Factoring the First Rational Expression

The numerator of the first rational expression can be factored as follows:

x2+9xy+8y2=(x+4y)(x+2y)x^2 + 9xy + 8y^2 = (x + 4y)(x + 2y)

The denominator of the first rational expression is already factored as 2xβˆ’6y=2(xβˆ’3y)2x - 6y = 2(x - 3y).

Factoring the Second Rational Expression

The numerator of the second rational expression is already factored as xβˆ’3yx - 3y. The denominator of the second rational expression can be factored as follows:

x2+6xyβˆ’16y2=(x+8y)(xβˆ’2y)x^2 + 6xy - 16y^2 = (x + 8y)(x - 2y)

Multiplying the Rational Expressions

Now that we have factored the numerator and denominator of each rational expression, we can multiply them together. We will start by multiplying the numerators and denominators separately.

Multiplying the Numerators

The numerator of the first rational expression is (x+4y)(x+2y)(x + 4y)(x + 2y), and the numerator of the second rational expression is xβˆ’3yx - 3y. We can multiply these two expressions together as follows:

(x+4y)(x+2y)(xβˆ’3y)=x(x+2y)(xβˆ’3y)+4y(x+2y)(xβˆ’3y)(x + 4y)(x + 2y)(x - 3y) = x(x + 2y)(x - 3y) + 4y(x + 2y)(x - 3y)

Expanding this expression, we get:

x(x2βˆ’3xy+2xyβˆ’6y2)+4y(x2βˆ’3xy+2xyβˆ’6y2)x(x^2 - 3xy + 2xy - 6y^2) + 4y(x^2 - 3xy + 2xy - 6y^2)

Simplifying this expression, we get:

x(x2βˆ’xyβˆ’6y2)+4y(x2βˆ’xyβˆ’6y2)x(x^2 - xy - 6y^2) + 4y(x^2 - xy - 6y^2)

Multiplying the Denominators

The denominator of the first rational expression is 2(xβˆ’3y)2(x - 3y), and the denominator of the second rational expression is (x+8y)(xβˆ’2y)(x + 8y)(x - 2y). We can multiply these two expressions together as follows:

2(xβˆ’3y)(x+8y)(xβˆ’2y)=2(x2+5xyβˆ’16y2)2(x - 3y)(x + 8y)(x - 2y) = 2(x^2 + 5xy - 16y^2)

Simplifying the Expression

Now that we have multiplied the numerators and denominators together, we can simplify the expression. We will start by simplifying the numerator.

Simplifying the Numerator

The numerator of the expression is x(x2βˆ’xyβˆ’6y2)+4y(x2βˆ’xyβˆ’6y2)x(x^2 - xy - 6y^2) + 4y(x^2 - xy - 6y^2). We can factor out the common term (x2βˆ’xyβˆ’6y2)(x^2 - xy - 6y^2) as follows:

x(x2βˆ’xyβˆ’6y2)+4y(x2βˆ’xyβˆ’6y2)=(x+4y)(x2βˆ’xyβˆ’6y2)x(x^2 - xy - 6y^2) + 4y(x^2 - xy - 6y^2) = (x + 4y)(x^2 - xy - 6y^2)

Simplifying the Denominator

The denominator of the expression is 2(x2+5xyβˆ’16y2)2(x^2 + 5xy - 16y^2). We can factor out the common term 22 as follows:

2(x2+5xyβˆ’16y2)=2(x2+5xyβˆ’16y2)2(x^2 + 5xy - 16y^2) = 2(x^2 + 5xy - 16y^2)

Final Simplification

Now that we have simplified the numerator and denominator, we can simplify the expression further. We will start by canceling out any common factors between the numerator and denominator.

Canceling Out Common Factors

The numerator of the expression is (x+4y)(x2βˆ’xyβˆ’6y2)(x + 4y)(x^2 - xy - 6y^2), and the denominator is 2(x2+5xyβˆ’16y2)2(x^2 + 5xy - 16y^2). We can factor out the common term (x2βˆ’xyβˆ’6y2)(x^2 - xy - 6y^2) from the numerator as follows:

(x+4y)(x2βˆ’xyβˆ’6y2)=(x+4y)(xβˆ’3y)(x+2y)(x + 4y)(x^2 - xy - 6y^2) = (x + 4y)(x - 3y)(x + 2y)

We can also factor out the common term (x+2y)(x + 2y) from the denominator as follows:

2(x2+5xyβˆ’16y2)=2(x+2y)(x+8y)(xβˆ’2y)2(x^2 + 5xy - 16y^2) = 2(x + 2y)(x + 8y)(x - 2y)

Now that we have factored out the common terms, we can cancel them out.

Canceling Out Common Terms

We can cancel out the common term (x+2y)(x + 2y) from the numerator and denominator as follows:

(x+4y)(xβˆ’3y)(x+2y)2(x+2y)(x+8y)(xβˆ’2y)=(x+4y)(xβˆ’3y)2(x+8y)(xβˆ’2y)\frac{(x + 4y)(x - 3y)(x + 2y)}{2(x + 2y)(x + 8y)(x - 2y)} = \frac{(x + 4y)(x - 3y)}{2(x + 8y)(x - 2y)}

We can also cancel out the common term (xβˆ’2y)(x - 2y) from the numerator and denominator as follows:

(x+4y)(xβˆ’3y)2(x+8y)(xβˆ’2y)=(x+4y)(xβˆ’3y)2(x+8y)(xβˆ’2y)\frac{(x + 4y)(x - 3y)}{2(x + 8y)(x - 2y)} = \frac{(x + 4y)(x - 3y)}{2(x + 8y)(x - 2y)}

Final Answer

The final answer is (x+4y)(xβˆ’3y)2(x+8y)(xβˆ’2y)\boxed{\frac{(x + 4y)(x - 3y)}{2(x + 8y)(x - 2y)}}.

Introduction

In our previous article, we multiplied the given expression and simplified our answer as much as possible. The expression given was a product of two rational expressions, and we used the rules of algebra to simplify it. In this article, we will answer some of the most frequently asked questions about the expression and its simplification.

Q&A

Q: What is the final answer to the expression?

A: The final answer to the expression is (x+4y)(xβˆ’3y)2(x+8y)(xβˆ’2y)\boxed{\frac{(x + 4y)(x - 3y)}{2(x + 8y)(x - 2y)}}.

Q: How did you simplify the expression?

A: We simplified the expression by factoring the numerator and denominator of each rational expression, and then multiplying them together. We also canceled out any common factors between the numerator and denominator.

Q: What is the numerator of the expression?

A: The numerator of the expression is (x+4y)(xβˆ’3y)(x + 4y)(x - 3y).

Q: What is the denominator of the expression?

A: The denominator of the expression is 2(x+8y)(xβˆ’2y)2(x + 8y)(x - 2y).

Q: Can you explain the steps to simplify the expression?

A: Yes, we can explain the steps to simplify the expression. We started by factoring the numerator and denominator of each rational expression. Then, we multiplied them together and canceled out any common factors between the numerator and denominator.

Q: What is the significance of the expression?

A: The expression is a product of two rational expressions, and it can be used to solve problems in algebra and other areas of mathematics.

Q: Can you provide more examples of simplifying expressions?

A: Yes, we can provide more examples of simplifying expressions. For example, consider the expression x2+4x+4x+2\frac{x^2 + 4x + 4}{x + 2}. We can simplify this expression by factoring the numerator and denominator, and then canceling out any common factors.

Q: How do you know when to cancel out common factors?

A: We know when to cancel out common factors when we have factored the numerator and denominator of each rational expression, and we have identified any common factors between the numerator and denominator.

Q: Can you explain the concept of factoring?

A: Yes, we can explain the concept of factoring. Factoring is the process of expressing an expression as a product of simpler expressions. For example, consider the expression x2+4x+4x^2 + 4x + 4. We can factor this expression as (x+2)(x+2)(x + 2)(x + 2).

Conclusion

In this article, we answered some of the most frequently asked questions about the expression and its simplification. We explained the steps to simplify the expression, and we provided more examples of simplifying expressions. We also explained the concept of factoring and how it is used to simplify expressions.

Additional Resources

Final Thoughts

Simplifying expressions is an important skill in algebra and other areas of mathematics. By understanding how to simplify expressions, we can solve problems more efficiently and effectively. We hope that this article has been helpful in explaining the concept of simplifying expressions and providing examples of how to do it.