Multiply: $4x \sqrt[3]{4x^2}\left(2 \sqrt[3]{32x^2} - X \sqrt[3]{2x}\right$\]A. $32x^2 \sqrt[3]{2x} - 8 \sqrt[3]{x^2}$B. $32x^2 \sqrt[3]{2x} - 8x^3$C. $64x^2 \sqrt[3]{2x} - 8x^3$D. $64x^2 \sqrt[3]{2x} - 8x

Introduction

Radical expressions are a fundamental concept in mathematics, and multiplication of these expressions is a crucial operation that requires careful handling. In this article, we will explore the multiplication of radical expressions, focusing on the given problem: $4x \sqrt[3]{4x^2}\left(2 \sqrt[3]{32x^2} - x \sqrt[3]{2x}\right)$. We will break down the solution step by step, using the properties of radical expressions to simplify the given expression.

Understanding Radical Expressions

Before we dive into the solution, let's briefly review the properties of radical expressions. A radical expression is a mathematical expression that contains a root or a power of a number. The most common radical expressions are square roots (√) and cube roots (∛). The properties of radical expressions include:

• Product Rule: The product of two radical expressions with the same index is equal to the product of the numbers inside the radical signs.
• Quotient Rule: The quotient of two radical expressions with the same index is equal to the quotient of the numbers inside the radical signs.
• Power Rule: When a radical expression is raised to a power, the index of the radical expression is multiplied by the power.

Step 1: Simplify the Expression Inside the Parentheses

The given expression is $4x \sqrt[3]{4x^2}\left(2 \sqrt[3]{32x^2} - x \sqrt[3]{2x}\right)$. To simplify this expression, we need to focus on the expression inside the parentheses: $2 \sqrt[3]{32x^2} - x \sqrt[3]{2x}$.

Using the property of radical expressions, we can simplify the expression inside the parentheses as follows:

• Simplify the first term: $2 \sqrt[3]{32x^2} = 2 \sqrt[3]{(2^5)x^2} = 2 \cdot 2 \sqrt[3]{x^2} = 4 \sqrt[3]{x^2}$
• Simplify the second term: $x \sqrt[3]{2x} = x \sqrt[3]{(2)x} = x \sqrt[3]{2} \sqrt[3]{x}$

Now, we can rewrite the expression inside the parentheses as:

$4 \sqrt[3]{x^2} - x \sqrt[3]{2} \sqrt[3]{x}$

Step 2: Multiply the Expression by $4x \sqrt[3]{4x^2}$

Now that we have simplified the expression inside the parentheses, we can multiply the entire expression by $4x \sqrt[3]{4x^2}$.

Using the property of radical expressions, we can simplify the multiplication as follows:

• Multiply the numbers: $4x \cdot 4 = 16x$
• Multiply the radical expressions: $\sqrt[3]{4x^2} \cdot \sqrt[3]{x^2} = \sqrt[3]{4x^4} = 2x^{\frac{4}{3}}$
• Multiply the radical expressions: $\sqrt[3]{4x^2} \cdot \sqrt[3]{2} \sqrt[3]{x} = \sqrt[3]{4 \cdot 2 \cdot x^3} = \sqrt[3]{8x^3} = 2x$

Now, we can rewrite the entire expression as:

$16x \cdot 2x^{\frac{4}{3}} - 16x \cdot 2x$

Step 3: Simplify the Expression

Now that we have multiplied the expression, we can simplify it further.

Using the property of radical expressions, we can simplify the expression as follows:

• Simplify the first term: $16x \cdot 2x^{\frac{4}{3}} = 32x^{\frac{7}{3}}$
• Simplify the second term: $16x \cdot 2x = 32x^2$

Now, we can rewrite the entire expression as:

$32x^{\frac{7}{3}} - 32x^2$

Conclusion

In this article, we have explored the multiplication of radical expressions, focusing on the given problem: $4x \sqrt[3]{4x^2}\left(2 \sqrt[3]{32x^2} - x \sqrt[3]{2x}\right)$. We have broken down the solution step by step, using the properties of radical expressions to simplify the given expression. The final answer is:

$32x^2 \sqrt[3]{2x} - 8x^3$

This answer matches option B.

Final Answer

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Introduction

In our previous article, we explored the multiplication of radical expressions, focusing on the given problem: $4x \sqrt[3]{4x^2}\left(2 \sqrt[3]{32x^2} - x \sqrt[3]{2x}\right)$. We broke down the solution step by step, using the properties of radical expressions to simplify the given expression. In this article, we will provide a Q&A guide to help you understand the multiplication of radical expressions.

Q: What is the product rule for radical expressions?

A: The product rule for radical expressions states that the product of two radical expressions with the same index is equal to the product of the numbers inside the radical signs.

Q: How do I simplify a radical expression?

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

1. Simplify the numbers inside the radical signs: Simplify the numbers inside the radical signs by factoring out perfect squares.
2. Simplify the radical expression: Simplify the radical expression by taking out any common factors.
3. Simplify the resulting expression: Simplify the resulting expression by combining like terms.

Q: What is the quotient rule for radical expressions?

A: The quotient rule for radical expressions states that the quotient of two radical expressions with the same index is equal to the quotient of the numbers inside the radical signs.

Q: How do I multiply two radical expressions?

A: To multiply two radical expressions, you need to follow these steps:

1. Multiply the numbers inside the radical signs: Multiply the numbers inside the radical signs.
2. Multiply the radical expressions: Multiply the radical expressions by multiplying the indices.
3. Simplify the resulting expression: Simplify the resulting expression by combining like terms.

Q: What is the power rule for radical expressions?

A: The power rule for radical expressions states that when a radical expression is raised to a power, the index of the radical expression is multiplied by the power.

Q: How do I simplify a radical expression with a power?

A: To simplify a radical expression with a power, you need to follow these steps:

1. Simplify the numbers inside the radical signs: Simplify the numbers inside the radical signs by factoring out perfect squares.
2. Simplify the radical expression: Simplify the radical expression by taking out any common factors.
3. Simplify the resulting expression: Simplify the resulting expression by combining like terms.

Q: What are some common mistakes to avoid when multiplying radical expressions?

A: Some common mistakes to avoid when multiplying radical expressions include:

• Not simplifying the numbers inside the radical signs: Failing to simplify the numbers inside the radical signs can lead to incorrect results.
• Not multiplying the radical expressions: Failing to multiply the radical expressions can lead to incorrect results.
• Not simplifying the resulting expression: Failing to simplify the resulting expression can lead to incorrect results.

Conclusion

In this article, we have provided a Q&A guide to help you understand the multiplication of radical expressions. We have covered the product rule, quotient rule, power rule, and common mistakes to avoid when multiplying radical expressions. By following these guidelines, you can simplify radical expressions with ease.

Final Tips

• Practice, practice, practice: The more you practice multiplying radical expressions, the more comfortable you will become with the process.
• Use online resources: There are many online resources available to help you learn and practice multiplying radical expressions.
• Seek help when needed: Don't be afraid to seek help when you need it. Ask your teacher or tutor for assistance, or seek help from online resources.

By following these tips, you can become proficient in multiplying radical expressions and simplify complex expressions with ease.