Multiply The Rational Expressions And Simplify The Result.1. \[$\frac{4x^2y^8}{x^5y^6z^2} \cdot \frac{x^2y}{20x^8} \cdot \frac{-1}{5x^2y^2}\$\]2. \[$\frac{81x^6}{y^4} \cdot \frac{x^2}{46x^5y}\$\]3.

Introduction

Rational expressions are a fundamental concept in algebra, and multiplying them is a crucial operation in simplifying complex expressions. In this article, we will explore the process of multiplying rational expressions and simplifying the result. We will use three examples to illustrate the steps involved in multiplying rational expressions and simplifying the result.

Example 1: Multiplying Rational Expressions

Let's consider the following rational expression:

$\frac{4x^2y^8}{x^5y^6z^2} \cdot \frac{x^2y}{20x^8} \cdot \frac{-1}{5x^2y^2}$

To multiply these rational expressions, we need to follow the rules of exponents and simplify the resulting expression.

Step 1: Multiply the Numerators

The first step in multiplying rational expressions is to multiply the numerators. In this case, we have:

$4x^2y^8 \cdot x^2y \cdot -1$

Using the rules of exponents, we can simplify this expression as follows:

$4x^2y^8 \cdot x^2y \cdot -1 = -4x^4y^9$

Step 2: Multiply the Denominators

Next, we need to multiply the denominators. In this case, we have:

$x^5y^6z^2 \cdot 20x^8 \cdot 5x^2y^2$

Using the rules of exponents, we can simplify this expression as follows:

$x^5y^6z^2 \cdot 20x^8 \cdot 5x^2y^2 = 100x^{15}y^8z^2$

Step 3: Simplify the Resulting Expression

Now that we have multiplied the numerators and denominators, we can simplify the resulting expression by dividing the numerator by the denominator.

$\frac{-4x^4y^9}{100x^{15}y^8z^2}$

Using the rules of exponents, we can simplify this expression as follows:

$\frac{-4x^4y^9}{100x^{15}y^8z^2} = \frac{-x^{-11}y}{10z^2}$

Final Answer

The final answer is $\boxed{\frac{-x^{-11}y}{10z^2}}$.

Example 2: Multiplying Rational Expressions

Let's consider the following rational expression:

$\frac{81x^6}{y^4} \cdot \frac{x^2}{46x^5y}$

To multiply these rational expressions, we need to follow the rules of exponents and simplify the resulting expression.

Step 1: Multiply the Numerators

The first step in multiplying rational expressions is to multiply the numerators. In this case, we have:

$81x^6 \cdot x^2$

Using the rules of exponents, we can simplify this expression as follows:

$81x^6 \cdot x^2 = 81x^8$

Step 2: Multiply the Denominators

Next, we need to multiply the denominators. In this case, we have:

$y^4 \cdot 46x^5y$

Using the rules of exponents, we can simplify this expression as follows:

$y^4 \cdot 46x^5y = 46x^5y^5$

Step 3: Simplify the Resulting Expression

Now that we have multiplied the numerators and denominators, we can simplify the resulting expression by dividing the numerator by the denominator.

$\frac{81x^8}{46x^5y^5}$

Using the rules of exponents, we can simplify this expression as follows:

$\frac{81x^8}{46x^5y^5} = \frac{81x^3}{46y^5}$

Final Answer

The final answer is $\boxed{\frac{81x^3}{46y^5}}$.

Example 3: Multiplying Rational Expressions

Let's consider the following rational expression:

$\frac{x^3y^2}{2z^3} \cdot \frac{3y^2}{4x^2z}$

To multiply these rational expressions, we need to follow the rules of exponents and simplify the resulting expression.

Step 1: Multiply the Numerators

The first step in multiplying rational expressions is to multiply the numerators. In this case, we have:

$x^3y^2 \cdot 3y^2$

Using the rules of exponents, we can simplify this expression as follows:

$x^3y^2 \cdot 3y^2 = 3x^3y^4$

Step 2: Multiply the Denominators

Next, we need to multiply the denominators. In this case, we have:

$2z^3 \cdot 4x^2z$

Using the rules of exponents, we can simplify this expression as follows:

$2z^3 \cdot 4x^2z = 8x^2z^4$

Step 3: Simplify the Resulting Expression

Now that we have multiplied the numerators and denominators, we can simplify the resulting expression by dividing the numerator by the denominator.

$\frac{3x^3y^4}{8x^2z^4}$

Using the rules of exponents, we can simplify this expression as follows:

$\frac{3x^3y^4}{8x^2z^4} = \frac{3xy^4}{8z^4}$

Final Answer

The final answer is $\boxed{\frac{3xy^4}{8z^4}}$.

Conclusion

Introduction

Multiplying rational expressions is a fundamental concept in algebra, and it can be a bit tricky to understand at first. In this article, we will answer some of the most frequently asked questions about multiplying rational expressions.

Q: What is a rational expression?

A: A rational expression is an expression that can be written in the form of a fraction, where the numerator and denominator are both polynomials.

Q: How do I multiply rational expressions?

A: To multiply rational expressions, you need to follow the rules of exponents and simplify the resulting expression. Here are the steps:

1. Multiply the numerators.
2. Multiply the denominators.
3. Simplify the resulting expression by dividing the numerator by the denominator.

Q: What are the rules of exponents?

A: The rules of exponents are as follows:

• When multiplying two numbers with the same base, add their exponents.
• When dividing two numbers with the same base, subtract their exponents.
• When raising a number to a power, multiply the exponent by the power.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to follow these steps:

1. Factor the numerator and denominator.
2. Cancel out any common factors.
3. Simplify the resulting expression.

Q: What is the difference between multiplying rational expressions and adding or subtracting rational expressions?

A: Multiplying rational expressions involves multiplying the numerators and denominators, while adding or subtracting rational expressions involves adding or subtracting the numerators and keeping the denominator the same.

Q: Can I multiply rational expressions with different denominators?

A: Yes, you can multiply rational expressions with different denominators. However, you need to find the least common multiple (LCM) of the denominators before multiplying.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you need to list the multiples of each number and find the smallest multiple that is common to both.

Q: Can I multiply rational expressions with negative numbers?

A: Yes, you can multiply rational expressions with negative numbers. However, you need to follow the rules of exponents and simplify the resulting expression.

Q: How do I multiply rational expressions with variables?

A: To multiply rational expressions with variables, you need to follow the rules of exponents and simplify the resulting expression.

Q: Can I multiply rational expressions with fractions?

A: Yes, you can multiply rational expressions with fractions. However, you need to follow the rules of exponents and simplify the resulting expression.

Conclusion

Multiplying rational expressions is a fundamental concept in algebra, and it can be a bit tricky to understand at first. However, by following the rules of exponents and simplifying the resulting expression, you can simplify rational expressions and make them easier to work with. In this article, we have answered some of the most frequently asked questions about multiplying rational expressions.

Frequently Asked Questions

• What is a rational expression?
• How do I multiply rational expressions?
• What are the rules of exponents?
• How do I simplify a rational expression?
• What is the difference between multiplying rational expressions and adding or subtracting rational expressions?
• Can I multiply rational expressions with different denominators?
• How do I find the LCM of two numbers?
• Can I multiply rational expressions with negative numbers?
• How do I multiply rational expressions with variables?
• Can I multiply rational expressions with fractions?

Resources

• Algebra Handbook
• Rational Expressions
• Multiplying Rational Expressions

Conclusion

Multiplying rational expressions is a fundamental concept in algebra, and it can be a bit tricky to understand at first. However, by following the rules of exponents and simplifying the resulting expression, you can simplify rational expressions and make them easier to work with. In this article, we have answered some of the most frequently asked questions about multiplying rational expressions.