Which Expression Is Equivalent To 10 X 3 3 X 2 \frac{\sqrt[3]{10x}}{3x^2} 3 X 2 3 10 X ​ ​ ?A. X ( 5 X 3 ) 3 \frac{x(\sqrt[3]{5x})}{3} 3 X ( 3 5 X ​ ) ​ B. 3 ( 5 X 3 ) X \frac{3(\sqrt[3]{5x})}{x} X 3 ( 3 5 X ​ ) ​ C. 5 3 3 X \frac{\sqrt[3]{5}}{3x} 3 X 3 5 ​ ​

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression $\frac{\sqrt[3]{10x}}{3x^2}$. We will examine the different options provided and determine which one is equivalent to the given expression.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a root or a power of a number. In the given expression, $\sqrt[3]{10x}$ represents the cube root of $10x$. The cube root of a number is a value that, when multiplied by itself three times, gives the original number.

Simplifying the Given Expression

To simplify the given expression, we need to apply the rules of exponents and radicals. We can start by factoring the numerator and denominator separately.

$\frac{\sqrt[3]{10x}}{3x^2}$

We can rewrite the numerator as:

$\sqrt[3]{10x} = \sqrt[3]{10} \cdot \sqrt[3]{x}$

Now, we can rewrite the expression as:

$\frac{\sqrt[3]{10} \cdot \sqrt[3]{x}}{3x^2}$

Next, we can simplify the denominator by factoring out $x^2$:

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

Now, we can rewrite the expression as:

$\frac{\sqrt[3]{10} \cdot \sqrt[3]{x}}{3x \cdot x}$

Evaluating the Options

Now that we have simplified the given expression, we can evaluate the options provided.

Option A: $\frac{x(\sqrt[3]{5x})}{3}$

To determine if this option is equivalent to the given expression, we need to simplify it.

$\frac{x(\sqrt[3]{5x})}{3}$

We can rewrite the numerator as:

$x \cdot \sqrt[3]{5x} = x \cdot \sqrt[3]{5} \cdot \sqrt[3]{x}$

Now, we can rewrite the expression as:

$\frac{x \cdot \sqrt[3]{5} \cdot \sqrt[3]{x}}{3}$

This expression is not equivalent to the given expression.

Option B: $\frac{3(\sqrt[3]{5x})}{x}$

To determine if this option is equivalent to the given expression, we need to simplify it.

$\frac{3(\sqrt[3]{5x})}{x}$

We can rewrite the numerator as:

$3 \cdot \sqrt[3]{5x} = 3 \cdot \sqrt[3]{5} \cdot \sqrt[3]{x}$

Now, we can rewrite the expression as:

$\frac{3 \cdot \sqrt[3]{5} \cdot \sqrt[3]{x}}{x}$

This expression is not equivalent to the given expression.

Option C: $\frac{\sqrt[3]{5}}{3x}$

To determine if this option is equivalent to the given expression, we need to simplify it.

$\frac{\sqrt[3]{5}}{3x}$

This expression is equivalent to the given expression.

Conclusion

In conclusion, the correct answer is Option C: $\frac{\sqrt[3]{5}}{3x}$. This expression is equivalent to the given expression $\frac{\sqrt[3]{10x}}{3x^2}$.

Final Answer

The final answer is $\boxed{C}$.

Additional Tips and Tricks

• When simplifying radical expressions, it's essential to apply the rules of exponents and radicals.
• Factoring the numerator and denominator separately can help simplify the expression.
• Be careful when rewriting the expression, as small mistakes can lead to incorrect answers.

Common Mistakes to Avoid

• Not applying the rules of exponents and radicals when simplifying radical expressions.
• Not factoring the numerator and denominator separately.
• Not being careful when rewriting the expression.

Real-World Applications

Radical expressions are used in various real-world applications, such as:

• Physics: Radical expressions are used to describe the motion of objects and the behavior of waves.
• Engineering: Radical expressions are used to design and analyze complex systems, such as bridges and buildings.
• Computer Science: Radical expressions are used in algorithms and data structures to solve complex problems.

Conclusion

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression $\frac{\sqrt[3]{10x}}{3x^2}$. We will examine the different options provided and determine which one is equivalent to the given expression.

Q&A: Simplifying Radical Expressions

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 the given expression, $\sqrt[3]{10x}$ represents the cube root of $10x$.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to apply the rules of exponents and radicals. You can start by factoring the numerator and denominator separately.

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.

Q: How do I determine if two radical expressions are equivalent?

A: To determine if two radical expressions are equivalent, you need to simplify both expressions and compare them.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is $\boxed{C}$, which is equivalent to the expression $\frac{\sqrt[3]{5}}{3x}$.

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

A: Some common mistakes to avoid when simplifying radical expressions include not applying the rules of exponents and radicals, not factoring the numerator and denominator separately, and not being careful when rewriting the expression.

Q: What are some real-world applications of radical expressions?

A: Radical expressions are used in various real-world applications, such as physics, engineering, and computer science.

Frequently Asked Questions

Q: How do I simplify a radical expression with multiple terms in the numerator?

A: To simplify a radical expression with multiple terms in the numerator, you need to factor out the greatest common factor (GCF) of the terms.

Q: How do I simplify a radical expression with a negative exponent?

A: To simplify a radical expression with a negative exponent, you need to rewrite the expression with a positive exponent and then simplify.

Q: How do I simplify a radical expression with a fraction in the numerator?

A: To simplify a radical expression with a fraction in the numerator, you need to simplify the fraction and then simplify the radical expression.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill to master in mathematics. By applying the rules of exponents and radicals, factoring the numerator and denominator separately, and being careful when rewriting the expression, we can simplify radical expressions and solve complex problems.

Additional Resources

• For more information on simplifying radical expressions, please refer to the following resources:

• Khan Academy: Simplifying Radical Expressions
• Mathway: Simplifying Radical Expressions
• Wolfram Alpha: Simplifying Radical Expressions

Final Answer

The final answer is $\boxed{C}$.

Common Mistakes to Avoid

• Not applying the rules of exponents and radicals when simplifying radical expressions.
• Not factoring the numerator and denominator separately.
• Not being careful when rewriting the expression.

Real-World Applications

Radical expressions are used in various real-world applications, such as:

• Physics: Radical expressions are used to describe the motion of objects and the behavior of waves.
• Engineering: Radical expressions are used to design and analyze complex systems, such as bridges and buildings.
• Computer Science: Radical expressions are used in algorithms and data structures to solve complex problems.