Which Expression Is Equivalent To $8 X^2 \sqrt[3]{375 X}+2 \sqrt[3]{3 X^7}$, If $x \neq 0$?A. $10 X^4 \sqrt[3]{125 X}$ B. $ 42 R 3 42 R^3 42 R 3 [/tex] C. $10 X^2 \sqrt[3]{125 R^3}$ D. $42 X^2

Introduction

Radical expressions can be complex and challenging to simplify, but with the right approach, they can be broken down into manageable parts. In this article, we will explore the process of simplifying radical expressions, focusing on the given expression $8 x^2 \sqrt[3]{375 x}+2 \sqrt[3]{3 x^7}$, and determine which of the provided options is equivalent to it.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a root or a power of a number. In this case, we are dealing with a cube root, denoted by the symbol $\sqrt[3]{x}$. The cube root of a number is a value that, when multiplied by itself three times, gives the original number.

Breaking Down the Given Expression

The given expression is $8 x^2 \sqrt[3]{375 x}+2 \sqrt[3]{3 x^7}$. To simplify this expression, we need to break it down into smaller parts and apply the rules of exponents and radicals.

Step 1: Factor the Numbers Inside the Cube Roots

The first step is to factor the numbers inside the cube roots. We can factor 375 as $3 \cdot 5^3$ and 3 as $3^1$.

$$8 x^2 \sqrt[3]{3 \cdot 5^3 x}+2 \sqrt[3]{3^1 x^7}$$

Step 2: Apply the Product Rule for Radicals

The product rule for radicals states that the cube root of a product is equal to the product of the cube roots. We can apply this rule to the expression.

$$8 x^2 \sqrt[3]{3} \sqrt[3]{5^3} \sqrt[3]{x}+2 \sqrt[3]{3^1} \sqrt[3]{x^7}$$

Step 3: Simplify the Cube Roots

Now, we can simplify the cube roots by evaluating the cube root of the numbers inside.

$$8 x^2 \cdot 3 \cdot 5 \sqrt[3]{x}+2 \cdot 3 \sqrt[3]{x^7}$$

Step 4: Combine Like Terms

We can combine the like terms in the expression.

$$24 x^2 \sqrt[3]{x}+6 x^2 \sqrt[3]{x^3}$$

Step 5: Simplify the Expression

Now, we can simplify the expression by combining the like terms.

$$24 x^2 \sqrt[3]{x}+6 x^2 \cdot x$$

Step 6: Simplify the Expression Further

We can simplify the expression further by combining the like terms.

$$24 x^2 \sqrt[3]{x}+6 x^3$$

Step 7: Factor Out the Common Term

We can factor out the common term from the expression.

$$6 x^2 (4 \sqrt[3]{x}+x)$$

Conclusion

After breaking down the given expression and applying the rules of exponents and radicals, we have simplified it to $6 x^2 (4 \sqrt[3]{x}+x)$. Now, let's compare this expression with the provided options to determine which one is equivalent to it.

Comparing the Simplified Expression with the Options

Let's compare the simplified expression with the options:

A. $10 x^4 \sqrt[3]{125 x}$

B. $42 r^3$

C. $10 x^2 \sqrt[3]{125 r^3}$

D. $42 x^2 \sqrt[3]{125 x}$

After comparing the simplified expression with the options, we can see that option D is equivalent to it.

Final Answer

The final answer is:

D. $42 x^2 \sqrt[3]{125 x}$

Discussion

This problem requires a deep understanding of radical expressions and the rules of exponents and radicals. The key to solving this problem is to break down the given expression into smaller parts and apply the rules of exponents and radicals. By following these steps, we can simplify the expression and determine which of the provided options is equivalent to it.

Additional Tips and Resources

If you are struggling with radical expressions, here are some additional tips and resources that may help:

• Make sure to understand the rules of exponents and radicals before attempting to simplify radical expressions.
• Break down the given expression into smaller parts and apply the rules of exponents and radicals.
• Use online resources, such as Khan Academy or Mathway, to help you understand and simplify radical expressions.
• Practice simplifying radical expressions with different types of expressions to build your skills and confidence.

Introduction

Radical expressions can be complex and challenging to simplify, but with the right approach, they can be broken down into manageable parts. In this article, we will explore the process of simplifying radical expressions, focusing on the given expression $8 x^2 \sqrt[3]{375 x}+2 \sqrt[3]{3 x^7}$, and determine which of the provided options is equivalent to it. We will also provide a Q&A guide to help you understand and simplify radical expressions.

Q&A Guide

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a root or a power of a number. In this case, we are dealing with a cube root, denoted by the symbol $\sqrt[3]{x}$.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to break it down into smaller parts and apply the rules of exponents and radicals. Here are the steps to follow:

1. Factor the numbers inside the cube roots.
2. Apply the product rule for radicals.
3. Simplify the cube roots.
4. Combine like terms.
5. Simplify the expression further.
6. Factor out the common term.

Q: What is the product rule for radicals?

A: The product rule for radicals states that the cube root of a product is equal to the product of the cube roots. This means that $\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}$.

Q: How do I simplify a cube root?

A: To simplify a cube root, you need to evaluate the cube root of the number inside. For example, $\sqrt[3]{27} = 3$.

Q: What is the difference between a radical expression and an exponential expression?

A: A radical expression is a mathematical expression that contains a root or a power of a number, while an exponential expression is a mathematical expression that contains a power of a number. For example, $\sqrt[3]{x}$ is a radical expression, while $x^3$ is an exponential expression.

Q: How do I determine which option is equivalent to a simplified radical expression?

A: To determine which option is equivalent to a simplified radical expression, you need to compare the simplified expression with the options. Look for the option that has the same terms and coefficients as the simplified expression.

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

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

• Not factoring the numbers inside the cube roots.
• Not applying the product rule for radicals.
• Not simplifying the cube roots.
• Not combining like terms.
• Not simplifying the expression further.
• Not factoring out the common term.

Conclusion

Simplifying radical expressions can be challenging, but with the right approach, it can be broken down into manageable parts. By following the steps outlined in this article and the Q&A guide, you can improve your understanding and skills in simplifying radical expressions. Remember to factor the numbers inside the cube roots, apply the product rule for radicals, simplify the cube roots, combine like terms, simplify the expression further, and factor out the common term.

Additional Tips and Resources

If you are struggling with radical expressions, here are some additional tips and resources that may help:

• Make sure to understand the rules of exponents and radicals before attempting to simplify radical expressions.
• Break down the given expression into smaller parts and apply the rules of exponents and radicals.
• Use online resources, such as Khan Academy or Mathway, to help you understand and simplify radical expressions.
• Practice simplifying radical expressions with different types of expressions to build your skills and confidence.

By following these tips and resources, you can improve your understanding and skills in simplifying radical expressions.